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Craftsmanship  in  the 

Teaching  of  Elementary 

Mathematics 


BLACKIE   &   SON   LIMITED 

50  Old  Bailey,  LONDON 
17  Stanhope  Street,  GLASGOW 

BLACKIE  &  SON  (INDIA)   LIMITED 
Warwick  House,  Fort  Street,  BOMBAY 

BLACKIE  &  SON  (CANADA)  LIMITED 
TORONTO 


Craftsmanship  in_the_ 
Teaching  of  Elementary 

^ -  - -O -• "        v 

Mathematics 


BY 


F.  W.  WESTAWAY 


Formerly  one  of  H.  M.  Inspectors  of  Secondary  Schools 
Author  of  "  Scientific  Method,  its  Philosophical  Basis  and  its  Modes  of  A 
"  Science  Teaching:  What  it  Was— What  it  Is — What  it  Might  E 
"  The  Endless  Quest:  3000  Years  of  Science  "  &c. 


BLACKIE  &  SON  LIMITED 

LONDON  AND  GLASGOW 


First  issued  1931 
Reprinted  1934,  *937 


Printed  in  Great  Britain  by  Blackie  &  Son,  Ltd.,  Glasgow 


"  Have  some  wine"  the  March  Hare  said  in  an 
encouraging  tone. 

Alice  looked  round  the  table,  but  there  was  nothing 
on  it  but  tea.  "  /  don't  see  any  wine"  she  remarked. 

"  There  isn't  any''  said  the  March  Hare. 

"  Then  it  wasn't  very  civil  of  you  to  offer  it" 
said  Alice  angrily. 


"  You  are  sad,"  the  White  Knight  said.  "  Let  me 
sing  you  a  song  to  comfort  you." 

"  Is  it  very  long?"  Alice  asked,  for  she  had  heard  a 
good  deal  of  poetry  that  day. 

"  It's  long"  said  the  Knight,  "  but  it's  very,  very 
beautiful.  Everybody  that  hears  me  sing  it — either  it 
brings  the  tears  into  their  eyes,  or  else " 

"  Or  else  what?"  said  Alice,  for  the  Knight  had 
made  a  pause. 

"  Or  else  it  doesn't,  you  know. — The  song  is  called 
'  WAYS  AND  MEANS  ',  but  that's  only  what  it's  called, 
you  know!" 

"  Well,  what  is  the  song,  then?"  said  Alice. 

"  /  was  coming  to  that,"  the  Knight  said.  "  The 
song  really  is  '  A-SITTING  ON  A  GATE  ':  and  the  tune's 
my  own  invention" 


BY  F.   W.  WESTAWAY 

SCIENCE  TEACHING 

What  it  Was—  What  it  Is 

What  it  Might  Be 
Second  Impression.    10s.  6d.  net 

"  Get  the  book  and  read  it;  it  is  the  best  thing  yet.  It  is  packed 
with  practical  advice  which  will  always  be  of  value." 

— Journal  of  Education. 

"  His  hook  w«ll  set  many  a  young  teacher  on  the  right  path,  and 
will  help  many  an  older  one  to  raise  his  performance  to  a  much 
higher  level  or  excellence." — Nature. 

"Reveals  on  every  page  the  zestful  interest  of  a  true  craftsman  in 
teaching  blended  with  informed  good  sense.  .  .  .  This  book  should  be 
read  by  all  headmasters  and  headmistresses  in  secondary  schools,  and 
it  is  worthy  to  be  studied  by  every  teacher  of  science.  If  its  counsels 
are  adopted  and  followed  we  shall  see  a  great  and  beneficent  change 
in  the  present  method  of  dealing  with  science  as  a  factor  in  education." 

— Education  Outlook. 

"  This  is  a  remarkable  book,  critical  and  stimulating,  the  product  of 
the  author's  long  experience  as  teacher,  headmaster,  and  H.M.I. 
.  .  .  comprehensive  in  scope  and  so  practical  that  it  will  be  a  most 
helpful  guide  to  the  beginner  and  an  inspiration  to  all.  We  recommend 
it  unreservedly  to  all  engaged  in  science  teaching  in  schools  and 
universities." — School  Science  Review. 


PREFACE 


When  asked  to  write  a  book  on  the  teaching  of  Elementary 
Mathematics,  I  felt  doubtful  as  to  the  avenue  by  which  the 
subject  might  be  best  approached.  During  the  present  cen- 
tury, the  general  "  policy  "  and  "  attitude  "  to  be  adopted  in 
mathematical  teaching  have  been  discussed  by  so  many 
authorities  that  there  seemed  very  little  new  to  say.  Finally 
I  decided  that  class-room  craftsmanship  might  be  made  a 
suitable  basis  of  treatment.  Thus  the  book  is  not  intended  for 
the  experienced  teacher  who  has  already  acquired  skill  in  his 
art,  but  for  the  still  struggling  beginner.  In  the  leading  schools, 
mathematical  craftsmanship  probably  leaves  little  to  be  desired, 
but  the  leaven  has  yet  to  work  its  way  into  the  mass. 

From  the  great  variety  of  topics  that  come  within  the 
ambit  of  the  various  mathematical  subjects,  I  have  selected 
for  treatment  those  which,  in  my  experience,  seem  to  give 
young  teachers  most  difficulty.  To  treat  all  topics  that  come 
within  the  daily  practice  of  mathematical  teachers  is  impossible; 
it  would  mean  writing  a  dozen  books  rather  than  one. 

I  have  sometimes  been  asked  if,  as  an  Inspector  pursuing 
the  same  daily  round  year  after  year,  decade  after  decade,  I 
am,  when  listening  to  lessons  in  mathematics,  ever  amused, 
ever  really  interested,  ever  inclined  to  be  severely  critical, 
ever  bored. 

vii 


viii  PREFACE 

Amused?  Yes;  for  instance,  when  a  young  master  tells  his 
boys  that  mathematics  is  by  far  the  most  important  subject 
they  learn,  inasmuch  as  it  is  the  only  one  that  leads  them  into 
the  region  of  "  pure  thought  ". 

Really  interested?  Yes,  every  day  of  my  life.  In  the  crafts- 
manship of  even  a  beginner  there  is  almost  always  some  element 
of  interest;  in  the  craftsmanship  of  a  really  skilful  mathe- 
matical teacher  there  is  to  me  always  a  veritable  joy.  I  never 
enter  a  classroom  without  hoping  to  find  something  which  will 
make  an  appeal,  and  I  am  not  often  disappointed.  Sometimes 
disappointed,  of  course;  unfortunately  not  all  mathematical 
teachers  have  come  down  from  heaven. 

Severely  critical?  Yes,  occasionally,  more  especially  at  the 
rather  slavish  adoption  of  certain  doubtful  forms  of  traditional 
procedure.  For  instance,  a  teacher  may  include  in  the  work 
of  the  bottom  "  Set  "  of  a  Form  the  Italian  method  of  division, 
well  knowing  that  two-thirds  of  the  boys  will  thenceforth 
always  get  their  sums  wrong.  Another  teacher  may  adopt 
"  standard  form  ",  not  because  he  has  examined  it  and  found 
it  to  be  good,  but  because  "  everybody  does  it  nowadays  ". 
Instead  of  saying,  "  I  thought  we  had  to  do  these  things," 
why  do  not  mathematical  teachers  hold  fast  to  the  faith  which 
is  really  in  them?  If  their  faith,  their  faith,  includes  the  Italian 
method,  standard  form,  and  the  score  of  other  doubtful  ex- 
pedients that  spread  like  measles  from  school  to  school,  I 
have  nothing  more  to  say. 

Bored?  Yes,  though  not  often.  The  petrifying  stuff  often 
doled  out  to  Sixth  Form  specialists,  the  everlasting  Series  and 
Progressions,  the  old  dodges  and  devices  and  bookwork  "proofs" 
ad  nauseam  in  preparation  for  scholarship  examinations,  all 
this  is  virtually  the  same  now  as  forty  years  ago.  True,  teachers 


PREFACE  ix 

are  not  much  to  blame  for  this.  Boys  have  to  be  prepared  for 
the  scholarship  examinations,  and  according  to  prescription. 
But  that  does  not  soothe  an  Inspector  who  has  to  listen  to  the 
same  thing  year  after  year,  and  I  admit  that,  with  Sixth  Form 
work,  sometimes  I  am  almost  bored  to  tears. 

If  I  had  to  pick  out  those  topics  which  in  the  classroom 
make  the  strongest  appeal  to  me,  I  should  include  (i)  Arith- 
metic to  six-,  seven-,  and  eight-year-olds,  when  well  taught; 
(ii)  Beginners'  geometry;  (iii)  Upper  Fourth  and  Lower  Fifth 
work  when  the  rather  more  advanced  topics  in  algebra,  geo- 
metry, trigonometry,  and  mechanics  are  being  taken  for  the  first 
time  (not  the  Upper  Fifth  and  its  revision  work);  (iv)  Upper 
Sixth  work  when  examinations  are  over  and  the  chief  mathe- 
matical master  really  has  a  chance  to  show  himself  as  a  master 
of  his  craft.  Sixth  and  Upper  Fifth  Form  work  often  savours 
too  much  of  the  examination  room  to  be  greatly  interesting: 
everything  is  excluded  that  does  not  pay.  But  inasmuch  as 
examination  success  is  a  question  of  bread  and  cheese  to  the 
boy,  the  teacher  is  really  on  the  horns  of  a  dilemma,  and  very 
naturally  he  prefers  to  transfix  himself  on  that  horn  that  brings 
him  the  less  pain. 

Why,  of  all  the  subjects  taught,  is  mathematics  the  least 
popular  in  girls'  schools?  and  whyjs  [t_the  one  subject  in 
which  the  man  in  the  street  feels  no  personal  interest? 

It  is  not  because  mathematics  is  difficult  to  teach.  My  own 
opinion  is  that  it  improbably  the  easiest  of  all  subjects  to  teach. 
When  it  is  taught  by  well-qualified  mathematicians,  and  when 
those  mathematicians  are  skilled  in  their  teaching  craft,  suc- 
cess seems  always  to  follow  as  a  matter  of  course,  in  girls' 
schools  equally  with  boys'.  The  failure  to  make  any  head- 
way, even  under  the  best  conditions,  on  the  part  of  a  small 


x  PREFACE 

proportion  of  boys  and  a  rather  larger  proportion  of  girls  is 
probably  due  to  a  natural  incapacity  for  the  subject.  Had 
I  my  own  way,  I  would  debar  any  teacher  from  teaching  even 
elementary  mathematics  who  had  not  taken  a  strong. -doge^  of 
the  calculus,  and  covered  a  fairly  extensive  field  of  advanced 
work  generally.  It  is  idle  to  expect  a  mathematical  teacher  to 
handle  even  elementary  mathematics  properly  unless  he  has 
begjx.through  the  mathematical  mill.  And  yet  I  have  heard 
a  Headmaster  say,  "  He  can  take  the  Lower  Form  mathe- 
matics all  right;  he  is  one  of  my  useful  men:  he  took  a  Third 
in  History." 

As  long  as  University  Scholarships  are  what  they  are,  so 
long  will  Sixth  Form  specialists*  work  proceed  on  present 
lines.  But  one  purpose  of  the  book  is  to  plead  for  considera- 
tion of  the  many  neglected  byways  in  mathematics  and  for 
their  inclusion  in  a  course  for  all  Sixth  Form  boys;  suggestions 
to  this  end  are  made  in  some  of  the  later  chapters.  We  want 
a  far  greater  number  of  ordinary  pupils  to  become  mathe- 
matically interested,  interested  in  such  a  way  that  the  interest 
will  be  permanent;  and  we  want  them  to  learn  to  think  mathe- 
matically, if  only  in  a  very  moderate  degree.  Why  do  ordinary 
pupils  shrivel  up  when  they  find  a  mathematician  in  their 
midst?  It  is  simply  that  they  are  afraid  of  his  cold  logic. 

There  is,  in  fact,  a  curious  popular  prejudice  against  mathe- 
maticians as  a  class.  It  probably  arises  from  the  fact  that  we 
*~~™_-.._ .~—  -  --•—.,  *  j 

are  not  a  nation  of  clear  thinkers,  and  we  dislike  the  few  amongst 
us  who  are.  Foreigners — at  least  the  French,  the  Germans, 
and  the  Italians — are  mathematically  much  keener  than  we 
are.  They  seem  to  become  immediately  interested  in  a 
topic  with  mathematical  associations,  whereas  we  turn  away 
from  it,  disinclined  to  take  part  in  a  discussion  demanding 


PREFACE  xi 

rigorous  logical  reasoning.  Competent  observers  agree  that 
this  is  in  no  small  measure  traceable  to  the  fact  that  our 
school  mathematics  has  not  been  of  a  type  to  leave  on  the 
minds  of  ordinary  pupils  impressions  of  permanent  interest. 

We^  have  driven  Euclid  out  of  Britain,  but  we  must  all 
admit  that  he  stood  as  a  model  of  honest  thinking,  and  we 
miss  him  sadly.  Were  he  to  come  back,  frankly  admitting  his 
failings  and  promising  reform,  not  a  few  mathematicians 
would  give  him  a  warm  welcome.  It  is  only  a  very  few  years 
since  I  heard  my  last  lesson  in  Euclid,  and  that,  curiously 
enough,  was  at  a  Preparatory  School.  It  was  a  pleasure  to 
hear  those  12-year-old  boys  promptly  naming  their  authority 
(e.g.  I,  32;  III,  21;  II,  11)  for  every  statement  they  made,  and  it 
was  exceedingly  difficult  to  improvise  the  necessary  frown  of 
disapproval.  Of  course  those  small  boys  did  not  understand 
much  of  what  they  were  doing,  and  of  actual  geometry  they 
knew  little.  But  in  spite  of  this  they  were  learning  to  think 
logically,  and  to  produce  good  authority  for  every  assertion 
they  made.  Our  modern  ways  are  doubtless  better  than  the 
old  ways,  but  when  we  emptied  the  bath,  why  did  we  throw 
out  the  baby? 

The  very  last  thing  I  desire  to  do  is  to  impose  on  teachers 
my  ideas  of  methods.  Anything  of  the  nature  of  a  standardized 
mgthpd  in  English  schools  is  unthinkable.  The  Board  of 
Education,  as  I  knew  it,  never  issued  decrees  in  matters  affect- 
ing the  faith  and  doctrines  of  our  educational  system;  it  con- 
fined itself  to  making  suggestions.  Admittedly,  however, 
democracy  has  now  come  to  stay,  and  its  unfortunate  though 
inevitable  tendency  to  standardize  everything  it  controls  may 
ultimately  prove  disastrous  to  all  originality^in  teaching  methods, 
and  reduce  the  past  high  average  ofjnitm^ve^agdjrf 


xii  PREFACE 

independence  in  schools.  Let^every;  teacher^strive  to  base  his 
methods.,  on  a  venturous  originality.  Let  him  resist  to  the 
death  all  attempts  of  all  bureaucrats  to  loosen  the  bonds  of 
obligation  to  his  art,  or  to  mar  his  craftsmanship. 

But  though  I  plead  for  originality  I  desire  to  utter  a  warn- 
ing against  a  too  ready  acceptance  of  any  new  system  or  method 
that  comes  along,  especially  if  it  is  astutely  advertised.  It  is 
perhaps  one  of  our  national  weaknesses  to  swallow  a  nostrum 
too  readily,  whether  it  be  a  new  patent  medicine  or  a  new  method 
of  teaching.  What  good  reason  have  we  for  thinking  that  a 
teacher  of  1931  is  a  more  effective  teacher  than  one  of  1881,  or 
for  that  matter  of  2000  years  ago?  What  is  there  in  method,  or 
in  personal  intelligence,  that  can  give  us  any  claim  to  be  better 
teachers,  better  teachers,  than  were  our  forefathers?  When 
a  new  method  is  announced,  especially  if  it  be  announced  with 
trumpets  and  shawms,  write  to  the  nearest  Professor  of 
Education,  and  more  likely  than  not  he  will  be  able  to  give 
you  the  exact  position  of  the  old  tomb  which  has  been  recently 
ransacked. 

A  method  which  is  outlined  in  a  lecture  or  in  a  book  is 
only  the  shadow  of  its  real  self.  A  method  is  not  a  piece  of 
statuary,  finished  ~and  unalterable,  but  is  an_^er^chajniging 
thing,  varying  with  the  genius  of  the  particular  teacher  who 
handles  it.  It  is  doubtful  wisdom  to  try  to  draw  a  sharp 
antithesis  between  good  methods  and  bad:  the  relative  values 
of  abstractions  are  invariably  difficult  to  assess.  The  true 
antithesis  is  between  eff^tiy^jfli^^^  The 

method  itself  counts  for  something,  but  what  counts  for  very 
much  more  is  the  life  that  the  craftsman  whenjictually  at  work 
breathes  into  it. 

The  regular  working  of  mathematical  exercises  is  essential, 


PREFACE  xiii 

for  the  sake  not  only  of  the  examination  day  that  looms  ahead 
but  also  for  illuminating  ideas  and  impressing  these  on  the 
mind.  Nevertheless,  the  working  of  exercises  tends  to  dominate 
ourwork  far  too  much,  and  to  consume  tiifte  that  might  far 
more  profitably  be  devoted  both  to  the  tilling  of  now  neglected 
ground  of  great  interest  and  to  the  serious  teaching  of  the 
most  fundamental  of  all  mathematical  notions,  namely,  those  of 
number,  function,  duality,  continuity,  homogeneity,  periodicity, 
limits,  and  so  forth.  If  boys  leave  school  without  a  clear  grasp 
of  such  fundamental  notions,  can  we  claim  that  their  mathe- 
matical training  has  been  more  than  a  thing  of  shreds  and 
patches?  I  plead  for  a  more  adequate  treatment  of  these  things. 
The  terms  "  Forms  "  and  "  Sets  "  I  have  used  in  accord- 
ance with  their  current  meaning.  The  average  age  of  each 
of  the  various  sections  of  Form  II,  III,  IV,  and  V  is  considered 
to  be  12+,  13+,  14+,  and  15+,  respectively,  the  units  figure 
of  the  age  representing  the  Form:  this  is  sufficiently  accurate 
for  all  practical  purposes.  The  age  range  of  Form  VI  is  taken 
to  be  16+  to  18+.  "  Sets  "  represent  the  redistributed  mathe- 
matical groups  within  any  particular  Form;  for  instance, 
100  boys  in  the  different  sections  of  Form  IV  might  be  re- 
distributed into  4  Sets,  a,  /?,  y,  and  S.  Admittedly  it  is  in  the 
lower  Sets  where  skilled  craftsmanship  is  most  necessary. 

On  reading  through  the  manuscript  I  find  that  I  have 
sometimes  addressed  the  teacher,  sometimes  the  boy,  rather 
colloquially  and  without  much  discrimination.  I  crave  the 
indulgence  of  my  readers  accordingly. 

Ait  teachers  of  mathematics  should  belong  to  the  Mathe- 
matical Association.  They  will  then  be  able  to  fraternize 


xiv  PREFACE 

periodically  with  the  best-known  and  most  successful  of  their 
fellow- workers.  The  Mathematical  Gazette  will  provide  them 
with  a  constant  succession  of  lucidly  written  practical  articles, 
of  hints  and  tips,  written  by  teachers  known,  by  reputation  if 
not  personally,  to  everybody  really  interested  in  mathematical 
education;  also  with  authoritative  reviews  of  new  mathematical 
books.  Members  may  borrow  books  from  the  Association's 
Library,  and  the  help  and  advice  of  specialists  are  always  to 
be  had  for  the  asking. 

In  writing  the  book  my  own  views  on  numerous  points 
have  been  checked  by  constant  reference  to  Professor  Sir 
Percy  Nunn's  Teaching  of  Algebra  and  its  two  companion 
volumes  of  Exercises,  books  I  have  not  hesitated  to  consult 
and  to  quote  from,  in  several  chapters.  I  am  old  enough  to 
remember  the  great  reputation  Professor  Nunn  enjoyed  as  a 
gifted  teacher  of  mathematics  when  he  was  an  assistant  master 
thirty  years  ago.  The  methods  he  advocates  are  methods  which 
have  been  amply  tested  and  found  to  be  sound  and  practical. 
His  book  deals  with  algebra  in  the  broadest  sense  and  gives 
advice  on  the  teaching  of  trigonometry,  the  calculus,  and 
numerous  other  subjects. 

F.  W.  W. 


CONTENTS 


CHAP, 
I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

XL 

XII. 

XIII. 

XIV. 

XV. 

XVI. 

XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 

XXIII. 

XXIV. 

XXV. 

XXVI. 


TEACHERS  AND  METHODS      ------ 

WHICH  METHOD:    THIS  OR  THAT?         - 

"  SUGGESTIONS  TO  TEACHERS  " 

ARITHMETIC:    THE  FIRST  FOUR  RULES  -         ... 
ARITHMETIC:    MONEY  ------- 

ARITHMETIC:    WEIGHTS  AND  MEASURES  - 
ARITHMETIC:    FACTORS  AND  MULTIPLES  - 

ARITHMETIC:   SIGNS,  SYMBOLS,  BRACKETS.    FIRST  NOTIONS 

OF  EQUATIONS 

ARITHMETIC:  VULGAR  FRACTIONS  -  -  -  -  - 
ARITHMETIC:  DECIMAL  FRACTIONS  -  ... 
POWERS  AND  ROOTS.  THE  A  B  C  OF  LOG- 


RATIO  AND  PROPORTION  - 
COMMERCIAL  ARITHMETIC 


ARITHMETIC: 
ARITHMS 

ARITHMETIC: 
ARITHMETIC: 
MENSURATION  -------- 

THE  BEGINNINGS  OP  ALGEBRA 

ALGEBRA:  EARLY  LINKS  WITH  ARITHMETIC  AND  GEOMETRY 

GRAPHS      -       •  - 

ALGEBRAIC  MANIPULATION   ------ 

ALGEBRAIC  EQUATIONS 

ELEMENTARY  GEOMETRY       ------ 

SOLID  GEOMETRY         ----... 

ORTHOGRAPHIC  PROJECTION 

RADIAL  PROJECTION     ....... 

MORE  ADVANCED  GEOMETRY          .... 

GEOMETRICAL  RIDERS  AND  THEIR  ANALYSIS     -        -        - 
PLANE  TRIGONOMETRY         ...... 


Page 
1 

16 
20 

23 
42, 
47 
54 

62 
67 

78 

94 
100 
109 
118 
122 
132 
137 
177 
205 
225 
287 
293 
300 
308 
319 


XVI 


CONTENTS 


CHAP.  Page 

XXVII.    SPHERICAL  TRIGONOMETRY      -        -        -        -        -  381 

XXVIII.    TOWARDS  DE  MOIVRE.     IMAGINARIES       -  387 

XXIX.     TOWARDS  THE  CALCULUS          -         -         -         -         -  401 

XXX.    THE  CALCULUS.     SOME  FUNDAMENTALS     -         -         -  421 

XXXI.    WAVE    MOTION.      HARMONIC    ANALYSIS.      TOWARDS 

FOURIER     --------  451 

XXXII.     MECHANICS 480 

XXXIII.  ASTRONOMY 497 

XXXIV.  GEOMETRICAL  OPTICS 506 

XXXV.     MAP  PROJECTION 517 

XXXVI.     STATISTICS 638 

XXXVII.     SIXTH  FORM  WORK 653 

XXXVIII.     HARMONIC  MOTION 558 

XXXIX.    THE  POLYHEDRA 672 

XL.      MATHEMATICS  IN  BIOLOGY 584 

XLI.    PROPORTION  AND  SYMMETRY  IN  ART        -        -        -  688 

XLII.    NUMBERS:    THEIR  UNEXPECTED  RELATIONS         -         -  694 

XLIII.    TIME  AND  THE  CALENDAR 614 

XLIV.     MATHEMATICAL  RECREATIONS 615 

XLV.      NON-EUCLIDEAN    GEOMETRY       -  -  -  -  -617 

XLVI.    THE  PHILOSOPHY  OF  MATHEMATICS           -         -         -  623 

XLVII.    NATIVE  GENIUS  AND  TRAINED  CAPACITY  -  639 

XLVIII.    THE  GREAT  MATHEMATICIANS  OF  HISTORY        -         -  643 

XLIX.    MATHEMATICS  FOR  GIRLS 644 

L.    THE  SCHOOL  MATHEMATICAL   LIBRARY  AND   EQUIP- 
MENT            646 

APPENDICES 
I.    A    QUESTIONNAIRE    FOR    YOUNG    MATHEMATICAL 

TEACHERS 663 

II.    NOTE  ON  AXES  NOTATION 657 

INDEX 659 


PLATE 


POLYHEDRAL  MODELS 


Facing 
Page 

-      584 


CHAPTER  I 

Teachers   and   Methods 

Mathematical  Knowledge 

"  That  man  is  one  of  the  finest  mathematicians  I  have 
ever  had  on  my  Staff,  but  as  a  teacher  he  is  no  good  at  all." 

"  Oh  yes,  he  can  teach  all  right.  He  can  hold  a  class  of  40 
boys  in  perfect  order  for  an  hour.  The  trouble  is  that  his 
knowledge  of  mathematics  is  so  superficial." 

These  criticisms  of  Head  Masters  are  not  altogether  un- 
common. A  Head  Master  naturally  looks  for  two  things  in 
members  of  his  Staff:  sound  knowledge  of  a  subject,  and  skill 
in  teaching  that  subject. 

Suppose  that  a  teacher  has  spent  three  or  four  years  at 
the  University,  has  obtained  the  coveted  First  in  mathematics, 
and  has  then  spent  a  year  in  the  Training  College  Department 
of  a  University  or  of  a  University  College.  Can  he  then  claim 
to  be  a  competent  mathematician  and  a  skilful  teacher?  The 
answer  is  no. 

The  knowledge  of  mathematics  obtained  in  a  four  years' 
University  course  forms  an  admirable  foundation  on  which  to 
build,  but  how  much  mathematics  can  be  learnt  in  so  short  a 
period  as  four  years?  At  the  end  of  that  time  it  is  a  common 
thing  for  University  students  hardly  to  have  touched  the  serious 
mathematics  of  physics,  of  chemistry,  of  engineering,  of  ma- 
chinery and  structures,  of  aviation,  of  astronomy,  of  biology, 
of  statistics,  or  to  have  mastered  more  than  the  barest  elements 
of  the  philosophy  of  the  subject.  Assuming  that  it  takes  a  year 
to  acquire  an  elementary  knowledge  of  each  of  the  applied 

(E291)  1  2 


2  CRAFTSMANSHIP  IN  MATHEMATICS 

subjects  just  named,  the  newly-fledged  graduate  has  still  in 
front  of  him  a  long  spell  of  hard  work  before  he  can  claim  to 
be  a  mathematician  in  the  broader  sense  of  the  term.  As  for 
the  philosophy  of  mathematics,  he  might  still  be  a  babe  in 
the  subject  after  five  years'  hard  reading.  Mathematics  touches 
life  at  so  many  points  that  the  all-round  training  of  a  mathe- 
matician is  a  very  long  business.  If  by  the  age  of  35  a  mathe- 
matician has  acquired  a  fair  general  knowledge  of  his  subject, 
he  has  done  well.  Consider  the  mathematics  of  physics  alone: 
what  a  vast  field!  The  field  is,  of  course,  ploughed  up  and 
sown  by  the  teachers  of  physics,  with  the  result  that  there 
is  always  a  rich  harvest  for  the  mathematical  staff  to  reap. 
Is  that  reaping  always  done? 

Skill  in  Teaching:    Training 

Then  as  to  teaching.  How  can  a  mathematical  Training 
College  student  claim  to  be  an  efficient  teacher  at  the  end 
of  his  training  year?  Skill  in  any  art  can  be  acquired  only  by 
much  practice,  and  the  art  of  teaching  is_a  particularly  diffi- 
cult art.  Although  mathematics  is  one  of  th^BaQioot,-  perhaps 
the  easiest,  of  all  subjects  to  teach,  it  is  a  rare  thing  for  a 
mathematical  teacher  to  be  able  to  feel  at  all  satisfied  with 
his  professional  skill  before  the  age  of  .30.  He  is  lucky  if 
other  people  adjudge  him  efficient  before  the  age  of  35.  Every 
mathematical  subject  is  full  of  teaching  problems.  Every 
one  of  these  problems  can  be  solved  in  a  variety  of  ways. 
Every  one  of  these  ways  is  worth  testing.  And  all  this  takes  a 
long  time.  As  for  lucidity  of  presentation — a  prime  necessity 
in  all  mathematical  teaching — that  is  in  itself  an  art  which 
half  a  life-time  does  not  seem  long  enough  to  perfect. 

I  have  often  been  asked  by  mathematical:  teachers  who  have 
not  been  formally  trained  what  good  they  would  have  done 
by  going  to  a  Training  College.  They  are  inclined  to  argue 
that  very  few  of  the  Training  Colleges  have  on  their  Staffs 
a  front-rank  mathematician,  and  that  therefore  such  Training 
Colleges  are  not  in  a  position  to  deal  with  the  subject  effec- 


TEACHERS  AND  METHODS  3 

tively;  that  even  if  the  Training  Colleges  are  able  to  impress 
into  their  service  members  of  the  mathematical  staff  of  the 
local  University,  little  real  help  is  obtainable,  for  "  although 
a  University  Professor  can  teach  me  mathematics,  he  cannot 
teach  me  how  to  teach  boys  mathematics  ".  The  general  con- 
tention is  not  without  its  points,  but  the  strictures  certainly 
do  not  apply  to  all  Training  Colleges,  as  trained  mathematical 
teachers  who  have  been  through  the  hands  of,  say,  Professor 
Sir  Percy  Nunn  are  the  first  to  admit. 

The  further  criticism  that  the  time  spent  on  Psychology  at 
the  Training  Colleges  is  "  absolutely  wasted  ",  since  "  it  has 
no  practical  value  in  the  solution  of  actual  classroom  prob- 
lems ",  is,  perhaps,  rather  more  justified.  It  is  possibly  true 
that  the  almost  useless  introspective  psychology  of  half  a 
century  ago  still  hangs  about  some  of  the  Training  Colleges; 
I  do  not  know.  But  young  teachers  should  make  themselves 
acquainted  with  the  valuable  experimental  work  which  is  now 
being  done  by  psychologists  all  over  the  world.  These  ex- 
periments are  often  based  on  masses  of  actual  data  derived 
from  the  classroom.  It  is  true  that  the  definite  results  obtained 
so  far  are  rather  patchy;  a  complete  body  of  psychological 
doctrine  has  yet  to  be  built  up  into  something  that  may  claim 
to  be  "  science  ".  But  no  teacher  can  afford  to  ignore  the 
work  that  has  been  done  and  is  being  done,  if  only  because 
such  a  large  part  of  it  has  a  very  close  bearing  on  present-day 
school  practice. 

One  broad  distinction  between  the  outlook  of  a  teachei 
who  has  been  through  a  Training  College  and  that  of  one 
who  has  not  is  that  the  trained  teacher  has  usually  had  knocked 
QUL.Qf_hini_the jprejudice  which  he  (very  naturally)  felt  for 
his  own  special  subject,  whether  mathematics,  classics,  or 
what  not.  He  has  learnt  that  in  the  Common  Room  he  will 
become  a^  member  of  a  community  regresenting^alj  the  subjects 
of  the  curriculum,  and  that  "  all  these  subjects,  not :  Jiis  own 
subject^alone,  answer  to  dee£-seated  neejs_ofjhe  human 
spirit,  all  of  tftern  es«^  in  the  great  stream  of 

movement  called^  civilization  ".     The  Training  College  does 


4  CRAFTSMANSHIP  IN  MATHEMATICS 

its  best,  of  course,  to  turn  out  competent  craftsmen,  but  it 
does  much  more.  It  leads  its  students  to  understand  the 
real  meaning  of  education  and  something  of  its  significance 
in  relation  to  the  many-sided  business  of  life.  It  shows  them 
how  much  wider  education  is  than  mere  teaching.  Moreover, 
the  students  are  day  by  day  in  contact  with  men  who  have 
reflected  deeply  upon  both  education  in  the  broader  sense 
and  teaching  in  the  technical  sense. 

In  a  good  Training  College,  the  embryo  mathematical 
teacher  is  taught  not  only  how  to  convey  to  his  pupils  a  know- 
ledge of  arithmetic,  algebra,  geometry,  and  the  rest,  but  also 
how  to  make  himself,  in  the  greatest  measure  possible,  an 
active  intellectual  adventurer  in  the  realms  of  number  and 
space,  how  to  follow  up  the  labours  of  the  great  masters  of 
mathematical  thought,  and  how  to  catch  something  of  their 
spirit  and  outlook.  He  is  encouraged  to  question  accepted 
mathematical  values,  and  to  inquire,  in  a  critical  spirit,  what 
parts  of  the  traditional  curriculum  are  really  vital  and  what 
parts  have  only  a  conventional  value.  He  is  made  aware  that 
many  of  the  textbooks  contain  a  considerable  amount  of  useless 
lumber,  and  he  is  taught:  how  to  discriminate  between  methods 
that  are  sound  and  methods  that  are  otherwise. 

Then  again,  in  a  good  Training  College  the  student  is 
able  to^obtain  expert  advice  on  every  kind  of  difficulty  he  may 
meet  with  in  his  teaching  practice.  At  what  stage,  for  example, 
should  "  intuition  "  work  in  geometry  give  way  to  rigorous 
proof?  How  can  the^best  approach  be  made  in  the  teaching 
of  ratio  and  proportion?  "How  is  the  theory  of  parallels  to 
be  treated?  The  Training  College  may  have  to  tell  him  frankly 
that  in  his  present  state  of  pupilage  he  is  probably  not  yet 
fitted  to  deal  with  the  theory  of  parallels  except  in  an  empirical 
way,  since  the  inherent  difficulties  of  the  theory  can  only  be 
grappled  with  after  a  prolonged  and  careful  study  of  the  re- 
searches of  modern  geometry;  only  then  will  he  be  in  a  position 
to  disentangle  logic  from  intuition,  and  so  be  able  to  devise  a 
treatment  suitable  for  Sixth  Form  boys.  Briefly,  the  Training 
College  will  point  out  quicksands,  as  well  as  firm  rock,  for  in 


TEACHERS  AND  METHODS  6 

mathematical  teaching  quicksands  abound.  A  teacher  who  is 
not  trained  will  often  not  only  get  into  the  quicksands  but  take 
his  boys  with  him. 

It  may  be  said  that  the  great  majority  of  Secondary  School 
teachers,  especially  teachers  in  the  Conference  *  schools,  are 
not  trained.  That  is  true,  and  the  result  has  often  been  that 
such  teachers  have  bought  their  first  few  years*  experience  at 
the ^xpense  of  their  boys.  But  not  always.  If  a  young  man 
straight  down  from  the  University  goes  as  a  Student  Teacher, 
that  is  as  an  observer  and  learner,  in  a  large  school  where 
mathematics  is  known  to  be  well  taught,  and  if  the  Head  of 
the  mathematical  department  has  sufficient  leisure  to  guide 
him  in  his  reading  and  to  act  as  his  professional  friend,  tjhtf 
year's  training  may  not  be  inferior  to  that  at  a  Training  College, 
Some  authorities  urge  that  it  may  be  superior,  inasmuch  as 
the  Student  Teacher  spends  the  year  with  real  practitioners 
rather  than  with  theoreticians.  I  do  not  attempt  to  decide 
this  question. 

Still  another  alternative — a  very  common  one — is  for  the 
embryo  teacher,  just  down  from  the  University,  to  join  the 
Staff  of  a  big  school,  to  admit  freely  that  he  is  a  neophyte,  and 
to  beg  for  all  the  help  and  criticism  he  can  obtain  from  his 
mathematical  colleagues.  Let  him  invite  his  seniors  to  come 
to  hear  him  teach,  and  to  criticize  him,  and  let  him  beg  the 
privilege  of  being  present  occasionally  at  their  lessons.  Let 
him  seek  their  advice  as  to  a  suitable  course  of  reading.  But 
let  him  not  think  that  he  is  a  teacher  sent  straight  from  heaven 
to  rectify  the  views  and  methods  of  the  old  fossils  of  25  and 
upwards  he  may  find  in  the  Common  Room:  that  way  lies 
trouble. 

One  thing  that  the  untrained  beginner  should  never  do 
is  to  join  the  Staff  of  a  small  school  where  he  has  to  undertake 
the  respcfnsfbffify  of  tR^whele  of  the  mathematical  teaching. 
In  the  first  place  it  is  not  fair  to  the  school;  the  boys  are  sent 
there  to  be  taught.  In  the  second  place  it  is  not  fair  to  himself, 

*The  term  "Public"  school  is  now  ambiguous,  and  is  better  not  used.     All 
grades  of  schools  are  either  "public"  or  "private". 


6  CRAFTSMANSHIP  IN  MATHEMATICS 

for  how  is  he  to  learn  his  job?  Why  not  open  an  office,  and 
set  up  as  a  Consulting  Civil  Engineer  instead?  He  is  just  as 
competent  to  do  the  one  thing  as  the  other. 

In  the  old  days,  all  would-be  craftsmen — joiners,  brick- 
layers, mechanics,  and  others — served  a  seven  years'  apprentice- 
ship, and  they  really  learned  their  business;  they  became  skilled 
craftsmen.  The  system  is  dead,  their  work  being  done  mostly 
by  machinery.  But  we  cannot  teach  by  machinery — yet;  and 
skilled  craftsmanship  in  teaching  can  be  acquired  only  by  a 
great  deal  of  practice. 

Conventional  Practice 

How  is  the  value  of  a  lesson  in  mathematics  to  be  assessed? 
I  do  not  refer  to  the  ordinary  things  in  which  every  teacher 
with  a  year's  experience  ought  to  be  reaching  efficiency — 
class-management,  discipline,  use  of  the  blackboard,  expertness 
in  questioning  and  in  dealing  with  answers,  and  so  forth — 
but  to  the  lesson  as  a  mathematical  lesson.  The  commonest 
fault  of  the  young  mathematical  teacher  is  that  he  talks  too 
much;  he  lectures,  and,  if  he  is  teaching  the  Sixth  Form,  he 
often  uses  his  University  notes.  It  takes  some  young  teachers 
a  long  time  to  learn  the  great  lesson  that  the  Jthing  that  matters 
jft.QSLJg.  not  what  Jthey  give  out  but  what  the  boys  take  in; 
that  their  work  is  teaching,  not  preaching.""""" 

Another  common  fault  of  young  mathematical  teachers, 
and  not  all  experienced  mathematical  teachers  are  guiltless  of 
it,  is  the  adoption  of  a  particular  method  because  it  is  mathe- 
matically neat,  the  sort  of  method  that  appeals  to  a  mathema- 
tician as  a  mathematician,  not  the  method  that  is  the  most 
suitable  for  demonstrating  a  particular  principle  or  teaching 
a  particular  rule  so  that  the  child  can  understand  it._  If  a 
mathematical  teacher  thinks  that  a  mathematician  is  listening 
to  him,  he  is  more  often  than  not  keener  to  reveal  his  know- 
ledge of  mathematics  than  to  exhibit  his  teaching  power 
But  the  observer's  object  is  usually  to  discover  what  the  boys 
are  learning,  and  to  assess  the  value  of  the  teaching;  and  the 


TEACHERS  AND  METHODS  7 

very  u  neatness  "  of  the  method  adopted  is  quite  likely  to  be 
the  cause  of  the  boys  learning  next  to  nothing.  To  be  effective, 
a  method  must  be  simple  and  be  clear  to  the  boys.  Mathe- 
matical rigour  may  thus  have  to  Be""sacrlftced,  though  the 
rigour  then  sacrificed  will  come  later. 

Again,  young  mathematical  teachers  are  ajrt  to  be  hide- 
bound b^coiwe^  .  Generation  after  generation  of  boys 
are  told,  for  example,  that,  in  an  expression  like  m  +  n  X  p, 
the  multiplication  sign  should  take  precedence  over  the  addition 
sign.  Why  should  it?  Doubtless  the  original  suggestion  is 
hidden  away  in  some  old  textbook,  but  it  has  been  consistently 
adopted  by  modern  writers  as  if  it  were  something  sacrosanct. 
Surely  if  the  signs  are  not  to  be  taken  in  their  natural  or3er 
from  left  to  right,  it  is  the  business  of  the  person  who  frames 
the  question  to  insert  the  necessary  brackets,  and  not  leave 
the  wretched  little  learner  to  do  it. 

Let  the  Head  of  a  mathematical  department  in  a  big  school 
remember  that  the  place  of  honour  for  himself  or  for  any 
exceptionally  gifted  member  of  his  Staff  isjn  the  lower  Forms. 
The  beginners'  geometry  is,  more  than  any  other  mathematical 
work,  in  need  of  skilful  teaching.  The  hackneyed  stuff  usually 
doneT  by  the  Sixth  Form  specialists  can  quite  well  be  taken 
by  a  youngster  just  down  from  the  University.  He  may  not 
be  able  to  teach,  but  he  is  mathematically  fresh,  and,  if  the 
specialists  in  the  Sixth  have  been  previously  well  trained, 
they  can  usually  take  in  the  new  mathematics  even  if  it  is 
rather  clumsily  presented  to  them. 

The  textbooks  mentioned  in  the  course  of  this  volume 
are  intende^Tm  the  main,  foLJhose  teachers  to  read, who  are 
technically  untrained.  The  object  is  not  to  recommend  this 
'Book "or  that  took  for  adoption  in  schools:  that  is  not  part  oi 
our  purpose  at  all.  The  object  is  to  suggest  a  book,  written 
by  some  skilful  teacher,  for  the  novice  to  read  right  through, 
critically.  He  should  ask  himself  why  the  writer  has  approached 
and  developed  the  subject  in  that  particular  way.  He  should 
then  read  a  second  book,  then  a  third,  and  so  on,  noting  the 
different  ways  of  approach  and  of  development,  and  the  different 


8  CRAFTSMANSHIP  IN  MATHEMATICS 

ways  in  which  different  teachers  do  things.  Then  he  should 
settle  down  and  evolve  methods  for  himself.  He  should  not, 
unless  in  exceptional  circumstances,  copy  another  teacher's 
method.  Let  his  methods  be  part  of  himself,  things  of  his 
owiT  elation,  things  for  which  he  has  an  affection  because 
they  are  his  own  children. 

Let  him  realize  that  methods  of  teaching  mathematics,  as 
of  teaching  other  subjects,  are  largely  conventional.  What  is  a 
"  best "  method,  and  how  is  it  to  be  determined?  Is  it  a  specially 
"  neat  "  method,  invented  by  some  clever  mathematician?  If 
so,  is  it  a  simple  method?  Is  it  productive  of  accuracy?  Here, 
psychology  teaches  us  a  little,  though  not  yet  very  much, 
and  to  say  that  one  method  is  "  better  "  than  another  is,  more 
often  than  not,  merely  to  express  a  personal  preference.  The 
teacher  should  always  ask  himself,  which  method  works  out 
best  in  practice!  Let  every  teacher  make  up  his  own  mind, 
and  not  be  led  away  either  by  the  textbooks  or  by  the  critics, 
though  the  textbooks  will  always  help,  and  the  critics,  if 
competent,  are  worth  listening  to.  But,  however  good  the 
books  and  however  competent  the  critics,  let  him  take  their 
help  and  advice  critically. 

Psychology  kas  k?lE£<Lus  £  great  deal  over  certain  points  in 
the  teaching^ oT arithmetic.  Experiments  have  been  directed 
marnly'to  discovering  which  of  possible  alternative  methods 
is  productive  of  greatest  accuracy  amongst  children.  If  such 
experiments  are  sufficiently  numerous  and  varied,  and  if  the 
results  of  the  tests  are  fairly  uniform,  we  may  feel  it  advisable 
to  consider  a  particular  method  favourably.  But  people  who 
experiment  in  this  way  must  set  out  with  an  entirely  unpre- 
judiced mind.  Results  that  are  not  arrived  at  objectively 
carry  no  weight. 

Whether  psychology  has  yet  succeeded  in  devising  con- 
vincing testsjpf  personal  matb^atkal^  ability,  I  am  un- 
certain. The  validity  of  some  of  the  criteria  used  has  been 
seriously  questioned  by  recognized  authorities.  Thejrelatipns 
between  mathematical  ability  and  general  "  intelligeace " 
have  certainly  not  been  clearly  determined.  We  have  probably 


TEACHERS  AND  METHODS  9 

all  met  highly  intelligent  men  with  keen  logical  powers  who 
were  no  good  at  all  at  mathematics,  and  have  known  brilliant 
mathematicians  whose  lack  of  general  intelligence  in  non- 
mathematical  affairs  was  amazing.  We  do  not  yet  really 
know  if  mathematical  ability  can  be  trained,  or  whether  it  is, 
so  to  speak,  a  fixed  quantity  at  birth.  The  deductions  we  can 
legitimately  draw  from  mere  examination  successes  are  by 
no  means  certain;  even  poor  mathematicians  may  become 
adepts  in  the  use  of  crammers'  dodges. 

The  "Dalton  Plan  " 

The  question  is  sometimes  asked,  what  is  the  Dalton 
plan  of  teaching,  and  can  it  be  made  to  apply  to  mathematics? 

The  plan  originated  in  America  ia  1920  and  has  since 
been  introduced  into  a  certain  number  of  English  schools. 
"  The  aim  is  to  provide  for  the  differences  encountered  in 
individual  pupils."  Class  teaching  as  such  is  abolished,  and 
gives  way  to  organized  private  study,  ijn^wWch^jhe_£U£il, 
not  the  teacher,  becomes  the  principal  and  responsible  agent. 
Instead  oFT  course  of  lessons  prescribed  according  to  a 
time-table,  an  "  assignment  "  of  work,  to  last  for  a  month, 
is  prepared  by  the  teacher.  The  whole  "  plan  "  hinges  on 
these  assignments.  The  month's  task  is  divided  into  four 
weekly  allotments,  which  are  further  subdivided  jnto  daily 
units.  Instead  of  working  to  a  time-table,  the  pupil  is  free 
to  work  at  whatever  subject  he  pleases.  The  rooms  are  no 
longer  "  classrooms ",  but  subject  rooms,  each  being  in 
charge  of  a  specialist  teacher  and  being  provided  with  the 
necessary  books  and  material.  The  pupils  move  freely  about 
from  room  to  room.  The  instructors  are  consulted  at  any 
time  by  any  pupil;  it  is  their  duty  to  advise  and  help  when- 
ever asked  to  do  so.  Conferences  and  collective  discussions 
are,  Jipwever,  arranged  at  specified  Tipurs. 
""""'  There  is  a  certain  amount  of  acceptable  opinion  in  favour 
of  the  plan  as  regards  subjects  like  English  and  History,  but 
as  regards  Mathematics,  Science,  and  Modern  Languages, 


10  CRAFTSMANSHIP   IN  MATHEMATICS 

the  balance  of  opinion  is  undoubtedly  against  it.  For  one 
thing,  theTmajority  of  mathematical  textbooks  are  unsuitable; 
they  do  not  demonstrate  and  elucidate  principles  simply 
enough  for  average  pupils  to  understand,  with  the  conse- 
quence that,  in  some  schools  which  are  working  on  Dalton 
lines,  formal  lessons  on  new  principles  precede  the  work  by 
assignments,  which,  for  mathematics,  are  not  much  more 
than  a  few  general  directions,  and  exercises  to  be  worked. 

In  short,  the  plan  does  not  at  present  seem  to  be  favoured 
very  much  by  the  majority  of  teachers.  A  teacher  who  adopts 
the  plan,  no  matter  what  his  subject,  must  be  prepared  for 
greatly  increased  personal  labour;  if  his  subject  is  mathe- 
matics, he  must  be  prepared  for  some  measure  of  disappoint- 
ment too.  On  the  other  hand,  the  able  mathematical  boy,  if 
given  a  free  hand,  with  just  occasional  help  when  difficulties 
are  serious,  seems  to  run  away  quickly  from  all  the  others. 
The  plan  seems  to  pay  with  Sixth  Form  specialists,  who  have 
been^weil  trained  up  to  the  Fifth.  Such  boys,  if  provided 
with  good  textbooks,  can,  with  very  little  formal  teaching  or 
other  help,  make  remarkably  rapid  progress. 

The  one  general  conclusion  that  seems  to  emerge  from 
Dalton  experiments  is  that  pupils  would  do  better  if  left  to 
wrestle  more  for  themselves,  anckthat  in  the  past  we  have  all 
tended  to  teach  too  much.  Although  thejplan  as  ajglan  is,  in 
the  estimation  of  not  a  few  good  judges,  rather_top  revolu- 
tionary for  gejrier_aJLadpptipn,  it  must,  on  the  other  hand,  be 
admitted  that  a  clever  teacher  who  loves  teaching  for  its  own 
sake  may  be  something  of  a  danger;  hejrnay  do  too  much 
of  the  thinking,  and  leave  the  boys  too  little  to  do  for  them- 

-    — 


No  boy  can  become  a  successful  mathematician  unless 
he  rights  hard  battles  on  his  own  behalf. 


TEACHERS  AND  METHODS  11 


Some  General  Principles 

The  last  statement  does  not  mean  that  mathematical 
teaching  is  not  necessary.  For  all  pupils  save  perhaps  the 
very  best,  it  is  fundamentally  necessary,  and  above  all  things 
the  teaching  must  be  clear. — Strive  day  by  day  to  make  the 
expression  of  your  meaning  ever  clearer.  Choose  your  words 
carefully  and  use  them  consistently.  Never  mind  the  correct 
formal  definitions  of  difficult  terms.  Use  a  term  over  and 
over  again  always  in  exactly  the  same  sense  but  associated 
in  different  ways  with  different  examples,  until  its  exact 
significance  imposes  itself  on  the  pupil's  mind.  It  is  merely 
a  question  of  the  child  continuing  to  learn  new  words  much 
in  the  same  way  as  he  learnt  the  stock  of  common  words 
which  are  already  in  his  possession.  His  mother  did  not 
define  for  him  as  a  baby  such  words  as  milk,  mamma,  toes, 
pussie,  sleep,  naughty,  yet  he  learnt  to  understand  their 
meaning  almost  before  he  could  walk. 

In  short,  dojiot  worry  beginners  with  formal  definitions, 
or  abstractions  of  any  other  sort.  Of  course,  almost  from 
the  first,  the  boy  makes  crude  use  of  ajl  sorts  Q.f  crudely 
acquired  abstract  terms,  for  in  his  enumeration  work  and  in 
his  early  quantitative  measurements,  which  he  has  always 
associated  with  concrete  objects,  intuition  and  guesses  have 
played  a  large  part.  But  the  mathematical  ideas  and  pro- 
cesses which  he  uses  for  solving  different  practical  problems 
gradually  become  clearer,  and  he  begins  to  see  interrelations 
between  principle  and  principle,  and  to  distinguish  those 
which  are  mutually  connected  from  those  which  are  inde- 
pendent. As  the  subject  proceeds,  it  tends  to  become  more 
abstract;  experience  grows;  and  the  teacher  has  to  choose 
his  own  time  for  stepping  in  and  exacting  greater  and  greater 
logical  rigour.  Below  the  Sixth  Form,  mathematics  is  essen- 
tially a  practical ^^  instrument,  not  a  subject  for  philosophic 
speculation.  Never  press  forward  formal  abstract  considera- 
tions until  '  ~~  "*•—-• 


12  CRAFTSMANSHIP  IN  MATHEMATICS 

What  is  the  use  of  discussing  even  with  Fifth  Form  pupils 
the  rival  merits  of  Euclid's  parallel  postulate  and  Playf air's 
alternative  version?  For  all  lower  and  middle  forms,  some 
such  statement  as,  "  lines  which  intersect  have  different 
directions;  lines  which  have  the  same  direction  do  not  meet 
but  are  parallel  ",  is  good  enough,  and  it  need  not  be  sub- 
jected to  criticism  until  the  Sixth.  Then,  criticism  is  desirable. 

The  organization  of  mathematical  work  in  a  large  school 
is  a  simple  matter;  between  the  Junior  Forms  and  the  Sixth 
there  may  be  4  blocks  of  3  or  4  Sets  each.  When  the  Set 
system  prevails,  gradation  is  easy.  Let  the  work  of  the  top 
Set  of  a  block  be  much  sterner  and  more  exacting  than  in 
the  bottom  Set,  and  do  not  attempt  to  include  in  the  work  of 
the  bottom  Set  all  the  subjects,  or  even  all  the  topics  of  a 
particular  subject,  that  are  allotted  to  the  better  Sets.  For 
instance,  all  top  Sets  will  learn  logarithms.  But  bottom  Sets? 
Why  should  they?  What  difficult  calculations  will  they  have 
to  engage  in  that  logarithms  will  really  help?  None  in  school, 
and  none  after  leaving  school.  Why  then  should  such  dull 
boys  be  made  to  waste  their  time  by  poring  over  the  pages 
of  a  numerical  lexicon  and  then  getting  their  sums  wrong 
instead  of  right?  It  is  unutterably  silly.  It  is  sometimes  done 
because  teachers  have  not  the  courage  to  say  what  they  really 
think. 

The  timid  teacher  may  be  inclined  to  argue,  "  but  how  are 
we  to  provide  for  the  boy  who  during  the  year  happens  to  be 
promoted  a  Set?"  That  is  certainly  a  real  problem  of  school 
organization  and  must  be  faced.  But  the  needs  of  the  occa- 
sional boy  must  not  be  catered  for  at  the  expense  of  a  whole 
class.  And,  after  all,  there  will  be  much  the  same  minimum 
of  work  for  the  various  Sets  within  a  block,  and  inter-block 
promotions  after  the  first  year  or  two  will  be  rare. 

Again,  suppose  that  somebody  comes  along  and  asks  if 
you  teach,  say,  Vectors.  If  you  do  not,  you  probably  have 
a  good  reason  for  it,  perhaps  because  Lord  Kelvin  himself 
poured  scorn  on  them.  In  that  case  do  not  hesitate  to  say  so. 
Holdfast  to  your  faith.  But  re-examine  the  grounds  of  your 


TEACHERS  AND  METHODS  13 

faith  from  time  to  time.  It  may  be  that  you  will  find  new 
arguments  in  favour  of  vectors,  arguments  which  will  induce 
you  to  revise  your  opinions.  And  so  with  scores  of  other 
things.  Keep  an  eye  on  your  defences,  but  remain  captain 
of  your  own  quarter-deck. 

Mathematical  Reasoning 

All  mathematical  teachers  should  reflect  carefully  on  the 
nature  of  mathematical  reasoning,  and  should  see  that  their 
pupils  are  made  more  and  more  conscious  of  what  constitutes 
mathematical  rigour.  Mathematical  reasoning  is  not,  as  com- 
monly supposed,  deductive  reasoning;  it  is  based  upon  an 
initial  analysis jrf jthjy^en.  and,  being  analytical,  is,jn  essence 
'  ~*  " '  e.  The  threads  of  the  web  once  disentangled,  synthesis 


begins,  and  the  solution  of  the  problem  is  set  out  in  deductive 
4yqgs.    We  arrange  our^  arguments  ded^ 

y  easily^jFollow.  up  the   chain  to  .oiir^  .final 


conclusion.  If  this  mere  setting  ouTVere  the  whole  story, 
how  simple  it  would  be!  Consider  this  syllogism,  in  form 
typically  Euclidean  and  deductive: 

Major  premiss:  All  professional  mathematicians  are 
muddle-headed. 

Minor  premiss:  The  writer  of  this  book  is  a  professional 
mathematician. 

Conclusion:  Therefore  the  writer  of  this  book  is  muddle- 
headed. 

Now  the  conclusion  is  quite  possibly  true,  and  it  is  cer- 
tainly the  correct  conclusion  to  be  drawn  from  the  two  pre- 
misses. But  both  the  major  and  the  minor  premisses  are 
false  (the  writer  of  this  book  is  not  a  professional  mathe- 
matician: heaven  forbid!  he  is  only  a  teacher),  and  therefore 
the  conclusion,  even  if  materially  true,  is  logically  absurd. 
In  fact  the  main  source  of  fallacious  reasoning  almost  always 
lies  in  false  premisses.  The  truth  of  the  conclusion  cannot 
be  more  true  than  the  truth  of  the  premisses,  and^  a  scrutiny 
and  a  rigorous  analysis  of  tfi^ap  is  tWefore  always  necessary. 


H  CRAFTSMANSHIP  IN  MATHEMATICS 

At  bottom,  all  reasoning  is  much  of  the  same  kind,  and 
it  usually  turns  on  the  truth  or  falsehood  of  the  premisses. 
Clear  thinking  is  thus  indispensable:  probabilities  have  to  be 
weighed,  irrelevant  details  discarded,  the  general  rules  accord- 
ing to  which  events  occur  have  to  be  divined,  hypotheses  have 
to  be  tested;  the  general  rules  once  established,  the  derivation 
of  particular  instances  from  them  is  a  simple  matter. 

But  in  elementary  mathematics  fer  beginners,  the_prQ£i§ipn 
of  concrete  particular  instances  comes  first  in  importance. 
In  the  handling  of  his  subject  in  the  classroom,  the  mathe- 
matical teacher  cannot  be  too  concrete.  As  the  boys  advance 
from  Form  to  Form,  they  will  gradually  begin  to  understand, 
and  in  the  Sixth  to  realize  fully,  that  the  solving  of  every 
mathematical  problem  consists  first  of  disentangling,  then  of 
setting  out  and  classifying,  then  of  tracing  similarities  and 
finding  possible  connecting  links,  then  of  linking  up  and 
generalizing;  in  other  words,  o£  analysis  followed  by  syn- 
thesis. Although  without  generality  there  is  no  reasoning, 
without  concreteness  there  is  neither  importance  nor  signifi- 
cance. But  in  schools  logical  rigour  is  a  thing  of  exceedingly 
slow  growth. — We  shall  return  to  the  question  of  mathematical 
reasoning  in  a  later  chapter. 

More  often  than  not,  present-clay  writers  of  standard  text- 
books in  mathematics  strain  after  both  ultra-precision  of 
statement  and  the  utmost  rigidity  of  proof.  But  any  attempt 
in  schools  to  be  perfectly  exact  all  at  once,  to  include  in  every 
statement  all  the  saving  clauses  and  limitations  that  can  be 
imagined,  inevitably  ends  in  failure.  This  is  where  the  be- 
ginner, untrained,  just  down  from  the  University,  so  often 
blunders.  He  is  inclined  to  argue  that,  unless  his  classroom 
logic  is  as  unassailable  as  that  of  his  University  Professor, 
his  work  will  be  open  to  serious  criticism.  The  work  of  even 
Sixth  Form  specialists  cannot  be  placed  on  an  unimpeachable 
logical  basis.  The_ degree  of  riggur_rtiiat»,caa.,h£.jixacted  at 
apyjrtage  must  necessarily ^depend on the  degree  of  intellectual 
development  of^ilie  pupil.  A  school  can  never  become  a 
place  for  mathematicaTasceticism, 


TEACHERS  AND  METHODS  15 

But,  as  boys  get  older,  they  should  be  encouraged  to  read 
their  own  textbooks  "  up  and  down,  backwards  and  forwards  ". 
In  their  study,  let  us  say,  of  the  calculus,  let  them  first  obtain 
an  insight  into  general  elementary  processes,  and  then  pro- 
ceed at  once  to  simple  applications.  Ample  practice  in 
differentiation  and  integration  is,  of  course,  necessary,  but 
the  study  of  geometrical  and  dynamical  applications  must  not 
be  unduly  delayed.  It  is  these  that  will  excite  interest,  and 
will  help  greatly  to  produce  an  appreciation  of  fundamental 
principles.  But  again  and  again  go  back  to  a  more  critical 
examination  of  those  principles.  The  applications  will  have 
taught  the  boys  a  great  deal  of  the  inner  meaning  of  the  pro- 
cesses, and  the  more  abstract  discussions  will  then  be  made 
much  easier  by  the  fact  that  the  learner  has  acquired  a  fair 
stock  of  more  or  less  concrete  ideas. 

The  Fostering  of  Mathematical  Interest 

The  general  standard  of  mathematical  attainments  in 
Sixth  Forms  is  now  reasonably  satisfactory,  and  entrants  at 
the  newer  universities  are  beginning  work  of  much  the  same 
grade  as  entrants  at  Cambridge.  But  though  Sixth  Form 
specialists  are  doing  solid  work  (of  a  very  restricted  type,  it 
must  be  added),  the  amount  of  mathematical  work  being  done 
by  all  the  other  pupils  who  have  obtained  the  School  Certifi- 
cate is,  as  a  rule,  slight,  too  slight  and  much  toojacademic  for 
the  fostering,,,  of  a  Jife-Jong  interest  in  the  subject.  Let  the 
younger  race  of  teachers  wake  up  to  this  important  fact,  and 
help  to  put  things  right.  We  shall  refer  to  this  point  again. 

Books  to  consult: 

1.  Didaktik  des  mathematischen  Unterrichts,  Alois  Hofler. 

2.  A  Study  of  Mathematical  Education,  Benchara  Branford. 


16  CRAFTSMANSHIP  IN  MATHEMATICS 

CHAPTER   II 

Which  Method:    This  or  That? 

Old  and  New.    Rational  and  Rule-of-thumb. 

An  intelligent  woman,  who  is  known  to  have  done  a  fail 
amount  of  mathematics  in  the  days  of  her  youth,  recently 
received  a  bill  for  8s.  7£rf.,  representing  the  cost  of  7  Ib.  6  oz. 
of  lamb.  She  was  "  sure  "  that  the  ounces  and  farthings  had 
been  included  merely  for  the  purpose  of  cheating  her,  and 
she  telephoned  to  the  butcher  to  know  the  price  of  the  meat 
per  Ib.  She  was  quite  unable  to  calculate  the  amount  for  her- 
self (Is.  2d.). 

A  well-known  Inspector  of  the  nineties  dictated  this  sum 
to  a  class  of  11-year-olds:  "  Take  one  million  ten  thousand 
and  one  from  ten  millions  one  thousand  one  hundred  ".  As 
might  be  expected,  hardly  any  children  had  the  sum  right. 
The  Inspector  looking  grieved,  the  Teacher  gently  asked  him 
if  he  would  himself  work  the  sum  on  the  blackboard.  Very 
unwisely  the  Inspector  tried  to  do  so,  and  made  a  hopeless 
mess  of  it — to  the  delight  of  the  boys. 

The  first  story  illustrates  ojiejDf  the  commonest  faults  of 
scBool  mathematics:  teachers  are  apt  to 'push  on  into  more 
advanced  work  before  foundations  have  been  well  and  truly 
laid.  The  second  story  shows  that  a^non-mathematician 
should  not  be  allowed  to  criticize  mathematical  teacher^.  To 
the  non-specialist,  mathematics  is  full  of  pitfalls,  and  it  may 
be  hoped  that  the  time  will  come  when  every  teacher  of  the 
subject  will  be  a  trained  mathematician,  even  if  he  has  to 
teach  nothing  but  elerrientary  arithmetic. 

Not  the  least  important  question  for  a  teacher  of  elemen- 
tary arithmetic  to  consider  is  the  method  ...of  ._aBPtflacb  ,  tQ .  a 
new  rule.  Should  that  rule  be  given  to  the  child  dogmatically, 
given  as  a  rule,  to  be  followed  by  the  working  of  examples 


WHICH  METHOD:    THIS  OR  THAT?  17 

until  it  is  thoroughly  assimilated?  or  should  the  rule  be 
"  explained  ",  approached  "  intelligently  ",  and  be  thoroughly 
"  understood  ",  before  it  is  applied  to  examples?  In  other 
words,  is  it  immoral  or  is  it  legitimate  to  provide  a  child  with 
a  working  tool  before  the  nature  of  the  tool  is  explained? 

To  put  it  another  way:  suppose  that  we  teach  a  rule 
"  intelligently  ",  and  the  children  get  50  per  cent  of  their 
sums  right;  or  suppose  that  we  teach  by  rule  of  thumb  and  the 
children  get  80  per  cent  of  their  sums  right.  Which  plan 
should  we  adopt? 

Should  we  give  credit  merely  for  "  getting  sums  right  "? 
or  should  we  forgive  mere  slips  if  the  working  shows  some 
grasp  of  the  process? 

Again:  suppose  we  find  that  some  of  the  newer  and 
popular  methods,  methods  that  have  superseded  those  in 
common  use  forty  or  fifty  years  ago,  are  less  productive  of 
speed  or  accuracy  or  both,  are  we,  or  are  we  not,  justified  in 
feeling  a  little  suspicious  of  the  newer  methods? 

Some  of  these  questions  have  been  answered  for  us  by  the 
psychologists,  who  in  recent  years  have  adopted  various 
devices  for  testing  the  comparative  merits  of  the  methods 
we  use  in  teaching  arithmetic.  The  old  school  of  psycholo- 
gists trusted  t&cKrnuch  to  intuition,  and  their  views  were 
doctrinaire. ^Present-day  psychologists,  on  the  other  hand, 
are  devoting  themselves  to  experiment,  to_the  garnering  of 
facts,  to  making  careful  deductions  from  those  facts.  For 
instance,  some  of  them  have**ar ranged  with  schools  for  tens 
of  thousands  of  simple  sums,  of  varying  types,  to  be  worked 
by  different  methods.  From  such  large  numbers  of  results 
legitimate  deductions  may  be  drawn,  especially  when  different 
psychologists  arrive  at  similar  conclusions  from  different  sets 
of  examples.  It  is  on  such  evidence  as  this  that  different 
methods  have  been  compared  and  some  sort  of  priority 
determined.  No  thoughtful  mathematical  teacher  would  now 
pronounce  dogmatically  in  favour  of.  his  own  method  of 
doi^  pithing,  even  if  he  has  used  it :  all  his  life. 'pHe  would 
suEJect  it,  ancl  other  methods  as  well,  to  prolonged  tests 

(B291) 


18  CRAFTSMANSHIP  IN  MATHEMATICS 

selecting  different  groups  or  classes  of  children  all  "  new  " 
to  the  principle  to  be  taught;  and  he  would  compare  the  re- 
sults in  different  ways,  for  instance  for  intelligence,  for 
accuracy,  and  for  speed;  and  he  would  make  sure  that  the 
general  conditions  of  the  tests,  for  instance  the  time  of  day 
when  they  were  given,  were  equalized  as  far  as  possible. — It 
is  in  such  matters  that  psychologists  are  helping  us  greatly. 

Perhaps  the  first  essential  of  all  is  accuracy,  especially 
Accuracy  in  all  kinds  of  computation.  What  would  be  the 
use  oT"a  bank  clerk  who^  made  mistakes  in  running  up  a 
column  of  figures?  A  tradesman  inaccurate  in  his  calculations 
might  jsoon  ^find  hunselt  a-bankrupt .  Indeed ,  accuracy jranjjs 
as  a  ^cardinal  viytiie^  it  Jgja  main  factor  of  morality.  A  boy 
who  gets  a^smnjw rongjshQ.uld~ be  made  to  get  it  right.  Never 
accept  a  wrong  answer.  This  does  not  mean  that  credit 
shbuldTTot  be  given  for  intelligence:  anything  but  that.  For 
instance,  a  boy  may  be  given  a  stiff  problem  and  get  it  wrong. 
But  that  problem  may  include  half  a  dozen  little  independent 
sums,  each  of  them  to  be  thought  out  before  it  can  be  actually 
worked;  five  of  them  may  be  right  and  one  wrong.  In  such 
a  complex  operation,  a  margin  of  error  may  be  legitimately 
allowed  for. 

If  we  think  of  our  own  personal  operations  in  arithmetic, 
those  we  are  engaged  in  day  by  day,  we  must  admit  that  most 
of  our  working  is  by  rule  of  thumb;  the  actual  rationale  of  a 
process  does  not  enter  our  heads.  We  have  become  almost 
mathematical  automata.  Yet,  if  called  upon  to  do  so,  we 
couIcCof  course,  explain  the  rationale  readily  enough.  But 
the  average  boy,  the  average  boy,  however  intelligently  he 
may  have  been  taught,  not^only  works  by  rule  of  thumb  but 
could  not  for  the  life  pf jiij&.give~an  adequate  explanation  pf 
tEe^ocess.  This  is  admittedly  brutal  fact.  Test  any  average 
class  of  30  boys,  twelve  months  after  they  have  been  taught 
a  new  rule,  and  it  is  highly  improbable  that  more  than  8 
or  10  will  explain  the  mathematical  operation  adequately 
and  intelligently.  The  experienced  teacher  never  expects  it. 

Nevertheless,  no  teacher  worth  his  salt  would  ever  dream 


WHICH  METHOD:    THIS  OR  THAT?  19 

q£  teaching  a  new  rule  j^ithout  %  approaching  _Jt  ratio.nally. 
He  would  do  his  best  to  justify  every  step  of  the  process, 
illustrating  and  explaining  as  simply  as  possible.  Perhaps 
4  or  5  of  the  boys  in  a  class  will  see  the  whole  thing  clearly, 
and  their  eyes  may  sparkle  with  satisfaction.  A  few  more, 
perhaps  8  or  10,  will  follow  the  argument  pretty  closely, 
though  if  asked  to  repeat  it  they  will  probably  bungle  pretty 
badly.  But  the  rest?  No.  They  want  the  rule,  simply  and 
crisply  put,  a  rule  they  can  follow,  a  rule  they  can  trust  and 
hold  fast  to.  And  no  teacher  need  break  his  heart  that  the 
majority  can  do  no  more.  No  inspector,  if  he  is  a  mathe- 
matician, ever  expects  more;  he  is  too  familiar  with  the 
mathematical  limitations  Jthat  nature  has  imposed  on  the 
average  boy;  of  British  origfn^  ~ 

As  a  boy  goes  up  the  school  and  his  intelligence  is  de- 
veloped, the  fundamental  processes  of  arithmetic  may  be 
made  clearer  to  him.  Any  average  boy  of  13  or  14  may  be  made 
to  understand  the  main  facts  of  our  decimal  system  of  nota- 
tion, whereas  at  7  or  8  he  may  have  failed  to  grasp  the  real 
significance  of  even  a  three-figure  number.  Every  teacher 
of  mathematics  should  remember  that  he  cannot  clear  the  ground 
finally  as  he  goes  along\  he  has  to  come  back  again  and  again. 

Do  not  worry  young  children  with  such  terms  as  abstract 
and  concrete.  Nothing  is  gained  by  telling  a  child  to  add  8 
sheep  to  9  sheep  instead  of  8  to  9.  Actual  arithmgtical  pro- 
cesse§jy*e^ll  abslrjcjt,  j^djyhe^jnotion.^f  ,  jcastingjsvjgry  sum 
into  problem  fprm_baL^J?^QQm^  a_§illy  fetish.  Present-day 
cEi!3ren  are  suffering  from  a  smjeUjjf  oranges  and^apples. 
Of  course  when  little  children  are  beginning  to  count,  to  add, 
to  subtract,  &c.,  the  use  of  real  things  is  essential,  and  in  this 
matter  we  may  learn  much  from  t  he  efficient 


teacfaer.    Some  of  the  very  fcest  arithmetic  J^achJn&J§  . 

school^.    It  is  a  pleasure  to  watch  chil- 


dren who  are  little  more  than  toddlers  getting  a  real  insight 
into  number  and  numeration.  The  worst  teaching  of  arith- 
metic I  have  ever  seen  was  in  the  lower  forms  of  the  old 
grammar  schools  of  40  years  ago.  In  those  days  it  was  not 


20  CRAFTSMANSHIP  IN  MATHEMATICS 

an  uncommon  thing  for  the  lowest  forms  to  be  placed  in 
charge  of  an  unqualified  hack.  Those  were  dark  days  indeed. 
"  Practical  "  mathematicsjincludes  manipulative  work  of 
some  kind,  actiiaf  measuring  as  well  as  calculating,  and  the 
more  of  this  irTtKT Seconds,  Third^  and  Fourths  the  better, 
especially  if  there  isTa  mathematical  laboratory  available. 
It  is  concrete  mathematics,  but  do  not  give  it  that  label.  In 
fact,  put  the  label  into  the  waste-paper  basket.  As  for  the 
label  abstract,  burn  it. 

Books  to  consult: 

1.  The  Approach  to  Teaching,  Ward  and  Roscoe. 

2.  The  New  Teaching  y  Adams. 

(These  are  not  books  specially  directed  to  mathematics,  but  to 
teaching  generally.  They  are  books  to  be  read  by  every  teacher, 
for  they  are  full  of  good  things.  Mr.  Ward  was  for  many 
years  chief  Inspector  of  Training  Colleges;  Mr.  Roscoe  is 
Secretary  to  the  Teachers'  Registration  Council,  and  was 
formerly  Lecturer  on  Education  at  the  University  of  Birming- 
ham; Sir  John  Adams  was  formerly  Professor  of  Education 
in  the  University  of  London.) 


CHAPTER  III 

"Suggestions   to   Teachers" 

The  Handbook  of  Suggestions  to  Teachers,  1928,  issued  by 
the  Board  of  Education,  contains  useful  hints  "  for  the  con- 
sideration of  teachers  and  others  concerned  in  the  work  of 
Public  Elementary  Schools  ".  The  practical  hand  is  revealed 
on  every  page,  and  there  can  be  no  doubt  that  the  best  teach- 
ing practice  known  in  the  country  is  embodied  in  it.  The 
book  deals  specifically  with  the  regui^ments jof Elementary 
Schools  as  they  are  likely  to  be  developed  during  thePnext 
few  years — Infant  ^Schools,  Junior  Schools,  and  Senior 


"SUGGESTIONS  TO  TEACHERS"  21 

Schools,  including  "  Selective  "  Central  Schools;  but  what 
is  said  about  mathematics,  especially  arithmetic,  is  equally 
applicable  to  scKoofs  oFaU  types. 

The  Board  are  of  opinion  that,  by  the  age  ofJLJU."  a  mini- 
mum course  should  at  least  include  a  thorough  groundwork 
in  notatigin,  a  knowledge  of  the  %s^J^r_rjjle^^^ied[  to 
mpngy,  and  the  ordinary  English  measures  of  length,  area, 
capacity,  weight,  and  jtime;  an  elementary  acquaintance  with 
vulgar  and  decimal  fractions,  together  with  simple  notions 
of  geom_etrigal  form  and  some  skill  in  practical  measure- 
ments." By  that  age,  "  accuracy  in  simple  operations  should 
in  great  measure  be  automatic.  It  depends  first  on  a  ready 
Knowledge  of  tables,,  and  secondly  upon  concentration,  but 
in  the  case  of  written  work  is  greatly  assisted  by  neatness  of 
figuring  and  clear  statement." 

(j^The  Board  contemplate  that,  in  future,  the  arithmetic  of 
all  Senior  Elementary  Schools  (where  the  age  will  extend 
from  11  to  14  or  15  or  even  16)  will  be  associated  with  men- 
Duration,  scale  ^drawing,  geometry,  gny>h§,  and  (for  boys) 
algebra,  trigonometry,  and  practical  mechanics.^  The  course 
of  mathematical  work  mapped  out  for  such  schools  is  par- 
ticularly suggestive  and  should  be  read  by  mathematical 
teachers  in  all  schools. 

The  Board  seem  also  to  contemplate  for  Senior  Elemen- 
tary Schools  some  form  of  mathematical  laboratory  where 
practical  work  can  be  done.  This  "work  is  to  be  associated 
with  the  geometry,  mensuration,  surveying,  mechanics,  and 
manual  instruction,  and  even  for  the  lqyv^r_cla§.s.es^.iiseful 
hints  are  given  for  practical. work  in  weights  and  nieasures. 
Much  of  the  work  which  at  one  time  constituted  the  pre- 
liminary course  of  practical  physics  might  be  included  as 
well — the  use  of  the  vernier  and  the  micrometer  screw  gauge, 
the  volumes  of  irregular  solids  by  displacement,  densities 
and  specific  gravities,  U-tube  work,  and  experimental  veri- 
fications of  such  principles  as  those  involved  in  the  lever 
and  pulleys,  in  the  pendulum,  and  in  Hooke's  law. 

The  time  has  gone  by  when  arithmetic,  even  in  Elemen- 


22  CRAFTSMANSHIP  IN  MATHEMATICS 

tary  Schools,  should  be  looked  upon  as  a  self-contained  sub- 
ject. Although  arithmetic  is  the  subject  dealing  with  numerical 
relations,  it  is  geometry  which  deals  with  space  relations,  and 
tfuTtwo  should  be  taught  together.  Algebra  is  just  a  useful 
mathematical  instrument,  full  of 


devices  for  both  arithmetic^and  geometry.  TTrigpnometry 
is  the  ^^  surveyor's  subject  ,jTTiseful  appl^itojp_or^j^bra,afld 
geometry  together.  A  graph  is  a  geometrical  picture,  showing 
arithmetical  amTalgebraic  relations  of  some  sort.  The  various 
subjects  fuse  tojjether  as  parts  of  a  single  puzzle^and  quite 
young  boys  may  be  given  a  working  insight  into  them  all. 
Arithmetic  alone  is  dry  bread  indeed,  far  too  beggarly  a 
mathematical  fare  even  for  a  Junior  School. 

From  the  first,  keep  the  mathematical  work  _jn_close  con- 
tact Ayith^thfe,  problems,  ^of  practical  life.  Let  matters  reasoned 
about  be  matters  with  which  the  children  are  either  already 
familiar  or  can  be  made  to  understand  clearly.  Po_  not,  take 
the  children  for  excursions  into  the  clouds^  what  is  perfectly 
cleaFlo  you  may  JBe^veiyJoggy  to  them.^  Therejs  no^  inde- 
pen3ent  "  jaculty  "  oTreagpning,  independent  of  the  par- 
ticular facts  and  relations  reasoned  about,  stored  away  ghost- 
like in  the  brain,  to  be  called  upon  when  wanted.  Hence, 
always  endeavour  to  ensure  that  the  things  which^  you  call 
upon  a  boy  to  discuss  are  seen  by  him  as  in  a  polished  mirror. 

of  mathematical  terms. 


If  you  take  care  always  to  use  such  terms  in  tHeir  exact  sense 
they  need  rarely  be  defined.  Even  very  small  children  have 
to  learn  the  terms  add,  subtract,  sum,  difference,  remainder, 
whole,  part,  less,  equal,  equals,  total,  and  older  children  m'ust 
acquire  an  exact  knowledge  of  such  terms  as  interest,  discount, 
gross,  net,  balance,  factor,  prime,  measure,  multiple,  and  dozens 
of  others.  If  you  use  them  consistently,  the  children  will 
soon  learn  to  appreciate  their  exact  significance. 

Let  part  of  your  stock-in-trade  be  price-lists  of  some  of 
the  big  London  stores,  the  Post  Office  Guide,  Bradshaw, 
Whitaker's  Almanack,  and  the  like.  Ask  yourself  what  sort  of 
mathematical  knowledge  the  children  are  likely  to  require 


"  SUGGESTIONS  TO  TEACHERS"  23 

inafter  life,  and,  as  far  as  you  canr  provide  accordingly.  But 
it_is  not  merely  a^guestion  of  giving  them  practicaT  tips: 
train  them  to  think  mathematically.  Train  them  to_car^_Jor 
accuracy.  Train  them  to  appreciate  some  of  the  marvels  of 
thelnirverse  —  the  very  great  and  the  very  small. 

^°  not  ^ise  .^Ijd-^sJuoned  methods  that  have  stood 


the  test  of  time,  and  do  not  be  too  ready  to  adopt  the  new- 
fangled methods  of  some  new  prophet.  Any  new  educational 
lubricant  which  is  advertised  to  be  a  tremendous  accelerator 
of  the  classroom  machinery  generally  proves  to  be  nasty 
clogging  stuff,  making  life  a  burden  for  those  who  use  it. 

Book  to  consult: 

Handbook  of  Suggestions  for  Teachers,  H.M.  Stationery  Office. 


CHAPTER   IV 

Arithmetic :    The   First   Four  Rules 

Numeration  and  Notation.    Addition 

We  have  already  mentioned  that  the  laborious  work  of 
psychologists  has  taught  us  much  about  the  pitfalls  experienced 
by  beginners  when  learning  arithmetic.  Few  young  teachers 
realize  the  number  of  separate  difficulties  felt  by  children  in 
learning  to  do  ordinary  addition  sums,  even  after  the  addition 
table  to  9  -f-  9  is  known. 

For  instance,  a  child  has  to  learn: 

(1)  To  keep  his  place  in  the  column; 

(2)  To  keep  in  mind  the  result  of  each  addition  until 

the  next  number  is  added  to  it;  and 

(3)  To  add  to  a  number  in  his  mind  a  new  number  he 

can  see; 


24  CRAFTSMANSHIP  IN  MATHEMATICS 

(4)  To  ignore  possible  empty  spaces  in  columns  to 

the  left; 

(5)  To  ignore  noughts  in  any  columns; 

(6)  To  write  the  figure  signifying  units  rather  than 

the  total  number  of  the  column,  specially  learn- 
ing to  write  0  when  the  sum  of  the  column  is 
20,  30,  &c.; 

(7)  To  carry. 

A  teacher  should  analyse  in  this  way  every  general  arithmetical 
operation,  and  provide  an  adequate  teaching  of  every  separate 
difficulty.  Unless  at  least  the  slower  pupils  are  thus  taught, 
they  may  break  down  in  quite  unsuspected  places.  Another 
important  thing  is  the  grading  of  difficulties.  For  instance, 
we  now  know  that  the  average  beginner  finds  the  addition  sum 

21  4 

43         easier  than         21 

35  3 

35 

and  the  latter  very  much  easier  than  21+  43  +•  35.  He  seems 
to  have  more  confidence  in  the  completed  columns ?  and  the 
vertical  arrangement  appeals  more  stronglyto  his  eye^  than 
does^T^horizonfal  arrangement 

Here  is  a  series  of  first  subtraction  sums,  graded  accord- 
ing to  the  difficulty  experienced  by  beginners: 

888 
380 

Teach  one  thing  at  a  time]  see  to  it  that  this  one  thing  does 
not  conceal  a  number  of  separate  difficulties;  and  let  that  one 
thing  be  taught  thoroughly  before  the  next  is  taken  up. 

The  bare  elements  of  numeration  and  notation  will  have 
been  taught  in  the  Infant  School  or  Kindergarten  School, 
and  on  entry  to  the  Junior  School  or  Department  the  children 
will  clearly  apprehend  the  inner  nature  of  a  3-figure  number, 
that,  for  instance, 

357  =  300  +  50  +  7. 


ARITHMETIC:    FIRST  FOUR  RULES 


25 


If  that  is  thoroughly  understood,  but  not  otherwise,  numeration 
and  notation  should  give  little  further  difficulty. 

Numbers  of  more  than  6  figures  will  seldom  be  required 
in  the  Junior  School  or  Department,  and  children  soon  learn 
to  write  down  6-figure  numbers  correctly.  Let  beginners 
have  two  3-column  ruled  spaces,  thus: 


Thousands. 

H. 

T. 

U. 

H. 

T. 

u. 

2 

4 

3 

5 

9 

6 

Tell  them  they  have  to  fill  up  the  spaces  under  "  thousands  " 
exactly  as  they  fill  up  the  old  familiar  3-column  space  on  the 
right.  Dictate  "  243  thousands  ",  and  pause;  the  child 
writes  243  under  "  thousands  ".  Now  go  on:  "  listen  to 
what  comes  after  thousands;  596  ".  The  child  soon  learns 
to  write  down  a  dictated  number  of  "  thousands  ",  just  as 
he  would  write  down  a  dictated  number  of  "  books  ".  With 
a  class  of  average  children  of  10  or  11  years  of  age,  one  lesson 
ought  to  be  enough  to  enable  them  to  write  down  even 
9-figure  numbers  accurately,  if  these  are  properly  dictated, 
and  if  the  children  are  first  made  to  understand  that  after 
"  millions  "  there  must  always  be  two  complete  groups  each 
of  3  figures,  the  first  of  these  groups  representing  thousands. 


Millions. 

Thousands. 

H. 

T. 

U. 

H. 

T. 

U. 

H. 

T. 

U. 

The  teacher  dictates:  "  Write  down  101  million  10  thousand 

and  one." 
"  How  many  millions?"     "  101."     "  Write  101  under 

millions'9 

"  How  many  thousands?"     "  10."     "  Write  10  under 
thousands" 


26  CRAFFSMANSHIP  IN  MATHEMATICS 

"  What  comes  after  thousands?"   "  1."   "  Write  1  in  the 
right-hand  3-column  space." 


1    0    1 


1    0 


"  Now  fill  up  with  noughts." 

Numbers  without  noughts  should  come  first.  Introduce  the 
noughts  gradually.  Remember  that  they  provide  constant 
pitfalls  for  beginners. 

Dispense  with  the  ruled  columns  as  soon  as  possible, 
but  let  the  successive  triads  always  be  separated  by  commas; 
14,702,116  (14  millions,  702  thousands,  116). 


Subtraction 

On  entry  to  the  Junior  School  the  child  will  already  have 
been  taught  that  the  subtraction  sum 


is  a  shortened  form  of 


800 
100 


867 
142 
725 

60 

40 


700  +  20  +  5 


bundles  of  sticks  or  bags  of  counters  being  used  to  make  the 
process  clear. 

They  will  also  have  been  taught  to  decompose  the  top 
line  in  such  a  sum  as  this,  leaving  the  lower  line  alone: 


534 
386 


Thus: 


500  +  30  +  4     __     400  +  120  +  14 

300  -j-  80  +  6     ~"     300  +80+6 

100+    40+8  =  148 


ARITHMETIC:    FIRST  FOUR  RULES  27 

I  have  often  seen  excellent  results  in  such  an  instance,  the 
small  children  handling  their  bundles  or  bags,  untying  them 
and  regrouping,  in  a  most  business-like  way.  They  really 
did  seem  to  have  grasped  the  essentials  of  the  process. 

But  can  the  method  be  regarded  as  the  most  suitable 
permanent  possession  for  older  children?    Consider  this  sum: 

80,003 
47,167 


The  necessary  decomposition  is  a  complicated  matter  for 
young  children.  They  have  to  take  10,000  from  the  80,000; 
leave  9000  of  the  10,000  in  the  thousands  column  and  carry 
1000  on;  leave  900  of  this  1000  in  the  hundreds  column  and 
carry  100  on;  leave  90  of  the  100  in  the  tens  column  and 
carry  10  on  to  the  units  column.  And  thus  we  have: 

80,000  +  3  70,000  +  9000  +  9004-90  +  13 

40,000  +  7000  +  100  +  60  +  7  40,000  +  7000  +  100  +  60+   7 

30,000*+2000  +  800  +  30+    6  =  32,836 

In  practice  this  is  what  we  see: 

7999 

**  %  q  q'3 
4  7,  1  6  7 

"~ 


I  confess  that,  judged  by  the  number  of  sums  right,  the  best 
results  I  have  ever  met  with  were  in  a  school  where  this 
decomposition  of  the  minuend  was  taught,  although  the 
teacher  responsible  was  not  only  not  a  mathematician  but  was 
entirely  ignorant  of  the  principles  underlying  the  plan  she 
had  adopted.  She  would  give  the  children  a  sum  like  this: 

70,024 
30,578 

Before  the  children  began  actually  to  subtract,  they  had 
to  examine  each  vertical  column  of  figures,  beginning  with 
the  units  column.  If  the  upper  figure  was  smaller  than  the 


28  CRAFTSMANSHIP  IN  MATHEMATICS 

lower,  they  would  borrow  1  from  "  next  door  ",  prefix  it 
to  the  unit  figure  in  the  top  line,  and  show  the  borrowing 
by  diminishing  by  1  the  figure  they  had  borrowed  from. 
Thus  they  wrote: 

l 

7  0,  0  *{14 

3  0,  5  7  8 

Then  they  would  examine  the  tens  column.  If,  as  before, 
they  found  the  upper  figure  smaller  than  the  lower,  they 
would  borrow  from  next  door  again — if  they  could;  if  not, 
they  would  pass  along  to  the  place  where  borrowing  was 
possible,  in  this  case  7,  change  the  7  to  6,  and  prefix  the 
borrowed  1  to  the  1  in  the  tens  column,  always  changing 
into  9's  the  O's  they  had  passed  over.  They  always  "  borrowed 
from  next  door  when  anybody  was  at  home,  putting  a  9  on 
the  door  of  every  house  they  found  empty  ".  Thus  the  sum 
was  made  to  look  like  this: 

6  9,  9  11 

X  q   ^   1^4 
3  0,  5   7   8 

Then  the  subtracting  was  begun,  and,  of  course,  it  was  all 
plain  sailing:  8  from  14,  7  from  11,  5  from  9,  0  from  9, 
3  from  6. 

Over  and  over  again  I  tested  that  class,  and  not  a  child 
had  a  sum  wrong.  But  the  children  had  no  idea  of  the  "  why  " 
of  the  process;  neither  had  the  teacher.  The  accuracy  was 
the  result  of  a  clear  understanding  of  an  exactly  stated  simple 
rule.  The  children  followed  the  rule  blindly. 

But  this  case  does  not  typify  my  general  experience, 
which  is  that  the  decomposition  method,  is  not  productive 
of  anything  like  the  accuracy  obtained  by  the  alternative 
method  of  equal  additions.  Although,  therefore,  I  am  drivgn 
to  favour  the  equal  additions  method,  this  method  does  not 
seem  quite  so  susceptible  of  simple  concrete  explanation 
for  very  young  beginners.  Still,  such  explanation  is  possible. 

First  try  to  make  the  pupils  understand  that  equal  additions 


ARITHMETIC:    FIRST   FOUR   RULES  29 

to  the  minuend  and  subtrahend  *  will  not  affect  the  difference. 
The  ages  of  two  children  provide  as  good  an  illustration  as 
anything: 

Jack  is  7  and  Jill  is  10;  their  difference  is  3.  In  4  years* 
time,  Jack  will  be  11  and  Jill  will  be  14;  their  difference 
will  still  be  3.  And  so  on.  Their  difference  will  always  be  3. 

A  first  lesson  on  brackets  will  serve  to  reinforce  the  idea: 

10-7  =3 

(10  -f  4)  -  (7  -f  4)  =  14  -  11  =  3 
(10  +  6)  -  (7  +  6)  =  16  -  13  =  3 
Then 

73  73  +  5  =  78  73  +  10  =  83 

21         or,         21  +  5-26          or,         21  -f  10  =  5U 
52  52  52 

In  this  way,  get  the  child  to  grasp  the  cardinal  fact  that  in 
any  subtraction  sum  we  may,  before  subtracting,  add  any- 
thing we  like  to  the  top  line  if  we  add  the  same  thing  to  the 
bottom  line;  the  answer  will  always  be  the  same.  Another 
example: 

73  70  4-  3  70  -^13 

48  40  +  8  50+8 

Here  we  have  added  10  to  the  top  line,  turning  3  into  13, 
and  we  have  added  10  to  the  bottom  line,  turning  40  into 
50.  (The  double  arrow  usefully  draws  attention  to  the  two 
additions.)  Thus  the  answer  to  the  altered  sum  will  be  the 
same  as  to  the  original  sum.  In  this  way  it  is  easy  to  give  a 
clear  understanding  of  the  so-called  "  borrowing  "  process. 

But  the  small  child  is  not  quite  so  happy  when  working 
by  this  method  in  the  concrete,  as  he  is  with  the  decomposition 
method.  When  he  is  given  the  two  extra  bundles  of  10, 
he  does  not  always  believe  that  the  sum  can  be  the  same. 

*  Do  not  use  these  terms  with  beginners;  wait  until  the  senior  school.  The /row 
line  and  take  line  will  do,  or  the  top  line  and  bottom  line,  or  the  upper  line  and  lower 
line.  Adopt  simple  terms  of  some  kind,  and  adhere  to  them  until  the  children  are 
better  prepared  to  adopt  a  stricter  nomenclature. 


30  CRAFTSMANSHIP  IN  MATHEMATICS 

However,   some   kindergarten   teachers   seem   to    have    little 
trouble  about  it. 

In  a  sum  like  the  following,  the  teaching  jargon  *  we 
should  use  for  beginners  would  probably  be  something  like 
this: 

"  Whenever  wejyjve_l  to  thetop  line,  we  musLalwapgive 

1  to  the  bottom  line  as  well,  but  next  door." 

9023 
3765 

"  5  from  3  we  cannot;  give  1  to  the  top  line  and  so  turn 
3  into  13;  5  from  13  is  8.  Now  give  1  to  the  bottom  line, 
next  door;  1  and  6  is  7." 

"  7  from  2  we  cannot;  give  1  to  the  top  line,  and  so  turn 

2  into  12;    7  from  12  is  5.    Now  give  1  to  the  bottom  line, 
next  door;    1  and  7  is  8." 

"  8  from  0  we  cannot;  give  1  to  the  top  line  and  so  turn 
0  into  10;  8  from  10  is  2.  Now  give  1  to  the  bottom  line, 
next  door;  1  and  3  is  4. 

"  4  from  9  is  5." 

The  words  borrow  and  pay  back  tend  to  mislead  the 
slower  boys,  since  we  borrow  from  one  line  and  pay  back 
to  another.  To  them  this  seems  unfair,  especially  when  we 
say  we  borrow  10  and  pay  back  only  1. 

Personally  I  prefer  to  give  1  to  the  top  line  and  never 
talk  about  paying  back,  but  compensate  by  giving  1  to  the 
bottom  line.  But  the  2  parts  of  each  double  transaction  must 
be  worked  in  association  at  once;  this  satisfies  the  children's 
sense  of  justice. 

In  each  of  the  first  several  lessons,  ask  what  the  giving 
of  1  really  signifies.  "  When  we  turned  3  into  13,  the  1  given 
was  really  10;  did  we  give  the  bottom  line  the  same  number? 

*  The  term  jargon  is  rather  suggestive  of  slang,  but,  of  course,  what  I  really  mean 
is  the  simple  homely  language  which  we  mathematical  teachers  all  invent  for  teaching 
small  boys,  language  which  rather  tends  to  offend  the  ear  of  the  English  purist.  But 
that  does  not  matter.  The  important  thing  is  to  express  ourselves  in  words  which 
convey  an  exact  meaning  to  the  children's  minds, 


ARITHMETIC:    FIRST  FOUR  RULES  31 

Yes,  because  the  6  which  by  adding  1  we  turned  into  7 
is  in  the  tens  column. 

"  When  we  turned  the  2  into  12,  the  2  was  really  20, 
and  the  1  we  gave  to  it  was  really  100;  did  we  give  the  bottom 
line  the  same  number?  Yes,  because  the  7  which  by  adding 
1  we  turned  into  8  is  in  the  hundreds  column." 

And  so  on.  A  very  £nnsui era r>le  proportion  of  the  children 
will  not  at  this  stageunderstand  the  process  at  all.  But  do 
not  worry  about  that.  Come  back  to  it  in  a  Ygsy^Jimg* 

It  will  be  weeks,  even  months,  before  the  slower  child 
will  have  had  enough  practice  to  do  subtraction  quickly  and 
accurately,  and  it  is  best  to  adhere  all  the  time  to  precisely 
the  same  form  of  teaching  "  jargon  ". 

There  remains  the  question,  shall  we  teach  the  children 
(i)  actually  to  subtract,  or  (ii),  to  add  (complementary  ad^- 
ditionj,  or  (iii),  first  to  subtract  from  10  and  then  add  the 
difference  to  the  figure  in  the  top  linel  For  instance, 

13 


Shall  we  say  5  from  13  is  8  i 
or,  shall  we  say  5  and  8  is  13?  ' 
or,  shall  we  say  5  from  10  is  5  and  3  is  8? 

The  last  must  be  ruled  out  of  court;  it  is  productive  of  great 
inaccuracy  amongst  beginners,  though  later  on  it  is  useful  in 
money  subtraction.  The  jsecond^Elan  is  popular,  but  it  has 
been  proved  to  be  less  productive  of  accuracy  than  thejiffi; 
and  it  is  Something  ot'  a  sham,  for  thenumber  to  be  added 
must  be  obtained  by  subtraction.*  Hence  the  first  method, 
honest  subtraction,  is  strongly  advocated,  and  that  demands 
ample  practice  in  both  the  subtraction  and  the  addition 
tables.  Thus  the  child  learns: 


both 


5  and  1  is  6 
5  and  2  is  7 
5  and  3  is  8 


and 


6  from  6  is  1 
6  from  7  is  2 
5  from  8  is  3 


&c. 


*  Cf .  algebraic  subtraction. 


32  CRAFTSMANSHIP  IN  MATHEMATICS 

"5  from  8?"  is  as  effective  a  form  of  question  as  (and  is 
much  more  elegant  than)  "  5  and  what  makes  8?". 

The  Tables 

The  child  must  learn  the  addition  table  to  9  +  9  perfectly. 
He  must  be  able  to  say  at  once  that,  e.g.,  9  and  8  is  17,  He 
must  also  be  able  to  say  at  once  that  9  froni^rLiS-A  and 
that  8jTom_jL7  is  9.  In  fact,  the  addition  and  subtraction 
tables  should  be  learnt  in  close  association.  Very  young 
children  when  learning  to  count,  to  add  and  subtract,  will, 
of  course,  be  shown  how  to  find  out  that  8  +  3—11  and 
that  11  —  3  =  8,  but  the  time  must  soon  come  when  they 
can  give  those  results  pat,  without  calculation  or  thinking 
of  any  kind;  and  this  means  a  great  deal  of  sheer  ding-dong 
work  from  which  lower  forms  and  classes  can  never  escape. 
Never  mind  the  charge  of  unintelligence;  be  assured  that 
the  people  who  make  such  a  silly  charge  have  never  had  to 
face  the  music  themselves.  Table  accuracy  is  the  one  key 
tp_ accurate  arithmetic. 

^EaclTsubtraction  table  is,  of  course,  as  already  indicated, 
the  mere  complement  of  an  addition  table.  For  instance, 
the  4  times  addition  table  begins  4  and  1  is  5  and  ends  4 
and  9  is  13;  the  corresponding  subtraction  table  begins 
4  from  5  is  1  and  ends  4  from  13  is  9.  Carry  the  addition 
tables  to  9  +  9  and  the  subtraction  tables  to  18  —  9. 

How  many  repetitions  are  necessary  to  ensure  permanent 
knowledge?  All  experienced  teachers  know  that  this  varies 
enormously.  It  may  be  that  only  10  repetitions  are  required, 
but  it  may  be  500,  according  to  the  individual.  Test,  test, 
test,  day  by  day.  Do  not  waste  the  time  of  a  whole  class 
because  further  drill  is  necessary  with  a  few. 

Helpful  blackboard  tests  may  be  given  in  a  variety  of 
forms,  e.g., 

9  +  6  =  *          17 -4  =  # 

6  +  8  =  x  13  +  8  =  x  &c. 

In  examples  of  this  kind  we  have  the  germ  of  equations, 


ARITHMETIC:    FIRST  FOUR  RULES  33 

as  we  had  with  the  examples  in  brackets.  Explain  that  x 
is  a  symbol  for  the  number  to  be  found.  Call  on  a  member 
of  the  class  and  point  to  the  first  x,  then  call  on  another  member 
and  point  to  the  second  x.  But  do  not  call  on  members  in 
order.  Keep  every  child  in  expectation.  Call  on  Smith  the 
shirker  half  a  dozen  times  a  minute.  If  the  answers  are  not 
given  at  once,  without  any  calculations,  the  tables  are  not 
known,  and  more  drill  is  necessary. 

Draw  a  circle  of  small  numbers  on  the  board  and  have 
them  added  together,  as  they  are  pointed  to.  The  answers 
must  be  instantaneous — or  the  tables  are  not  known. 

Make  the  children  count  forwards  and  backwards,  by 
1's,  then  by  2's,  then  by  3's,  &c. 

1,  4,  7,  10,  13,  16,  &c. 
100,  96,  92,  88,  84,  80,  &c. 

This  sort  of  practice  helps  the  tables  greatly. 

But  do  not  expect  that,  because  a  boy  knows  7  +  6  is  13, 
he  will  therefore  know  that  27  +  6  =  33.  Such  extended 
examples  require  special  practice,  and  the  practice  must 
be  continued  day  by  day  until  the  boy  knows  at  once  that  a 
7  added  to  a  6  always  produces  a  3.  Similarly  with  sub- 
traction; a  boy  must  be  able  to  say  at  once  that  a  7  taken  from 
a  6  always  produces  a  9. 

Write  on  the  blackboard,  say,  a  7.  "  Let  us  add  6's." 
Smith?  13;  Brown?  19;  Jones?  25;  dodging  about  the  class. 
The  response  must  be  instant.  Similarly  with  subtraction. 

Let  your  schemes  for  testing  the  tables  be  as  varied  as 
possible.  Do  not  be  satisfied  as  long  as  there  is  a  single 
mistake.  Do  not  forget  that  dull  boys  may  require  10  times, 
perhaps  50  times,  the  practice  that  quick  boys  require.  There 
must  be  no  counting  on  fingers,  no  strokes,  no  calculations  of 
ainy  kindT 


So  with  the  multiplication  and  division  tables.  Beginners 
are  taught,  of  course,  that  multiplication  is  just  a  shortened 
form  Q|  a  succession  of  additions,  and  dm8*011  a  _ shortened 
fofm^of  a  succession  of  subtractions.  That  fact  grasped, 

fB~29n "  " — 4 


34  CRAFTSMANSHIP  IN  MATHEMATICS 

then  come  the  tables,  multiplication  to  9  X  9  and  division 
to  81  -f-  9. 

Do  not  be  intelligently  silly  and  teach  a  boy  "  to  find 
out  for  himself  "  the  value  of  9  X  8  by  making  him  set  out 
9  rows  of  8  sticks  each  and  then  count  to  discover  72.  Make 
him  learn  that  9  X  8  ==72. _  When  he  begins  multiplication 
and  division,  a  few  very  easy  concrete  examples  will  be  given 
him,  to  make  the  fundamental  ideas  clear.  Then  make  him 
learn  his  tables,  learn  his  tables. 

As  with  the  addition  and  subtraction  tables,  write  the 
multiplication  and  division  tables  side  by  side.  The  sign 
for  "  equals  "  may  well  be  substituted  for  "  is  ". 

1X7=7  7's  into    7  =  1 

2  X  7  =  14  7>s  into  14  =  2 

3  X  7  =  21  7's  into  21  =  3 


9  X  7  =  63  7's  into  63  =  9 

Mental  work: 


Seven  threes? 
Three  sevens? 
Sevens  into  twenty-one? 
Threes  into  twenty-one? 


Let  the  3,  the  7,  and  the  21 
hang  together  in  all  4 
ways. 

And  so  on. 


Ask  for  the  factors  of  such  numbers  as  42,  77,  28,  &c. 

Blackboard  Work: 

Write  down  a  number  consisting  of  15  or  18  figures, 
and  ask  the  class  to  give  the  products  of  successive  pairs  of 
figures,  as  rapidly  as  possible:  e.g. 

371498652498. 
Answers: 

21,  7,  4,  36,  72,  &c., 
Again: 

7  X  3  =  x\  3  X  7  =  x\  3  X  x  =  21;  7  X  x  =  21;  x  X  3  =  21; 

x  X  7  =  21. 

21  21 

—  =  x\  —  =  x, 

37 


ARITHMETIC:    FIRST  FOUR  RULES  35 

And  so  on.  Point  to  an  #,  and  call  on  a  particular  pupil  for 
the  answer. 

Mental  work  in  preparation  for  multiplication  and  division 
sums: 

(3  x  7)  +  1  =  x.       (3  x  7)  +  2  =  x         (8  X  7)  +  5  -  x. 
3's  into  22  =  x.          3's  into  23  =  x.  8's  into  61  =  x. 

Ample  practice  in  this  type  of  example  is  necessary.  The 
examples  are  of  course  one  step  beyond  the  simple  tables; 
there  are  two  operations,  one  in  multiplication  or  division, 
one  in  addition  or  subtraction.  Hence  instantaneous  response 
is  hardly  to  be  expected  from  slower  children.  But  it  is 
surprising  how  quickly  the  answers  come  from  children 
who  know  their  tables,  who  know  that  8  X  7  =  56  and  that 
56  +  5  =  61,  though  it  is  well  to  remember  that  a  mental 
effort  is  required  to  keep  in  mind  the  first  answer  while  it 
is  being  further  increased  or  diminished. 

The  11  times  table  is  hardly  worth  learning.  The  12  times 
table  may^  be  ^ostponed^  until  money  sums  are  taken  up. 
The  15  times  is  easy  to  learn  and  is  useful  for  angle  division. 
So  is  the  20  times  table.  Mental  work  on  simple  multiples 
is  easy  to  provide,  e.g.  18  X  9  =  twice  9x9;  (16  X  7)  = 
twice  (8x7). 

But  when  actually  teaching  the  tables,  it  is  a  safe  rule 
not  to  complicate  matters  by  giving  tips  for  exceptional  cases. 
Do  not,  for  example,  tell  a  beginner  that,  when  he  is  adding 
a  column  of  figures,  he  should  look  ahead  to  see  if  two  of 
them  added  together  make  10.  If  he  has  to  find  the  sum  of 
4,  8,  3,  7,  5,  teach  him  to  say,  4,  12,  15,  22,  Jj7^not  to  look 
ahead  and  to  discover  that  3  +  7  =  107~and  then  to  say 
4,  12,  22,  27.  Such  a  plan  with  beginners  makes  for  in- 
accuracy. Good  honest  straightforward  table  work  must 
come  first.  Short  cuts  may  come  later,  when  they  may  be 
more  readily  assimilated. 


36  CRAFTSMANSHIP  IN  MATHEMATICS 


Multiplication 

It  is  easy  to  make  any  average  child  who  is  well  grounded 
in  numeration  and  notation  understand  that  4  times  273 
means  the  sum  of  four  273's,  i.e. 

273 
273 
273 
273 

and  that  therefore  the  answer  is 

(4  times  200)  +  (4  times  70)  +  (4  times  3); 

and   he   sees   readily  enough   that  the  teacher's  shortened 
form 

273  200  70  3 

4  444 

1092  =      800+280+12 

But  the  slower  child  will  not  understand,  though  he  will 
learn  the  ordinary  rule  of  multiplication  fairly  readily. 

In  teaching  multiplication,  the  advisable  succession  of 
steps  seems  to  be: 

(a)  Easy  numbers  by  2,  3,  and  4;   no  carrying;   no  zeros 
in  multiplicand. 

(b)  Easy  numbers  by  2,  3,  and  4;   no  carrying;   zeros  in 
multiplicand. 

(c)  Easy  numbers  by  2,  3  ...  9,  with  carrying;   no  zeros 
in  multiplicand. 

(d)  Easy  numbers  by  2,  3  ...  9,  with  carrying;  zeros  in 
multiplicand. 

(e)  The  same  with  larger  multiplicands. 
(/)  Multiplication  by  10. 

(g)  Multiplication  by  2-figure  numbers  not  ending  in  a 
zero. 


ARITHMETIC:    FIRST  FOUR  RULES  37 

(h)  Multiplication  by  2-figure  numbers  ending  in  a  zero, 
(i)  Multiplication  by  3-figure  numbers,  zeros  varied. 

Be  especially  careful  to  show  clearly  the  effect  of  multiplying 
by  10,  viz.  the  shifting  of  every  figure  in  the  multiplicand  one 
place  to  the  left  in  the  notational  scheme,  i.e.  each  figure  is 
made  to  occupy  the  next-door  position  of  greater  importance. 
Then  show  the  effect  of  multiplying  by  100,  by  1000,  by 
20,  200,  6000,  &c.  Bear  in  mind  that  the  work  has  particular 
value,  inasmuch  as  ultimately  it  will  lead  on  to  decimals. 

From  the  outset,  use  the  term  multiplier  and  the  term 
product,  but  let  the  difficult  term  multiplicand  wait  until  the 
senior  school  stage.  The  term  top  line  will  do  for  juniors. 

When  we  come  to  ordinary  2-figure  and  3-figure  multi- 
pliers, which  of  the  following  processes  is  preferable,  the 
first  or  the  second? 

34261  34261 

43  43 


102783  1370440 

137044  102783 


1473223  1473223 

The  first  is  the  old-fashioned  method;  the  second  is  newer  and 
at  present  is  popular.  The  second  is  often  advocated  because 
(1)  it  leads  on  more  naturally  to  the  rational  multiplication 
of  decimals,  (2)  it  is  preferable  to  multiply  by  the  more 
important  figure  first,  if  only  because  the  first  partial  product 
is  a  rough  approximation  to  the  whole  product. 

The  first  reason  does  not  appeal  to  me  at  all,  for  I  am 
very  doubtful  about  the  allied  method  of  multiplication  of 
decimals.  The  second  reason  is  undoubtedly  a  good  one. 

Numerous  tests  of  the  comparative  merits  of  the  two 
methods  have  shown  that  the  old  method  leads  to  a  much 
greater  accuracy  than  fE¥  new,  and  to  me  that  seems  greatly 
to  outweiglTthe  advantage  of  the  new  method.  Slower  boys 
seem  to  have  much  more  confidence  in  a  method  where  they 
have  to  begin  with  both  units  figures,  as  they  do  in  addition 


38  CRAFTSMANSHIP  IN  MATHEMATICS 

and  subtraction.    In  any  case  I  deny  that  the  newer  method 
is  "  more  intelligent  "  than  the  old7 


Division 

Begin  by  instructing  the  children  to  write  down  in  standard 
division  form  such  little  division  sums  as  they  know  from 
their  tables.  Teach  them  the  terms  dividend,  divisor,  and 
quotient:  we  can  hardly  do  without  them. 

2(6  3\9          3|7  5|9 

3  3  2,  and  1  over         1,  and  4  over. 

Now  teach  them  the  use  of  the  term  remainder,  and  to  write 
the  letter  R  for  it. 

5|9  4|6 

T,  R  4         I,  R  2 

Now  2-figure  dividends,  within  the  tables  they  know. 

4|36  8|47  9|79 

^9  •£,  R  7       J£,  R  7 

Now  2-figure  dividends  beyond  the  tables  they  know. 


"  4's  into  93?  the  tables  do  not  tell  us.   Then  let  us  take  our 
4  times  table  further: 

10  x  4  =  40 

11  X  4  =  44 


23  x  4  =  92 

24  X  4  =  96 


"  Evidently  4's  into  93  are  23,  and  1  R.   Hence 

4|93 
23,  R  1 


ARITHMETIC:    FIRST  FOUR  RULES  39 

"  But  we  need  not  have  written  out  that  long  table;  we  may 
work  in  this  way: 

"  4's  into  9?  2  and  1  over;  write  down  the  2. 

"  By  the  side  of  the  1  over,  write  down  the  3,  to  make  13. 

"  4's  into  13?  3,  and  1  over.   Write  down  the  3. 

"  The  last  1  over  is  our  Remainder. 

"  But  what  does  this  mean?  When  we  said  4's  into  9  we 
really  meant  4's  into  90,  and  when  we  wrote  down  the  2, 
the  2  really  meant  20.  Here  is  a  better  way  of  showing  it 
all,  and  we  will  write  the  figures  of  the  answers  above  the 
dividend,  instead  of  below  it. 


4)93 

8_  =  80  =  20  times  4 
13 
12  =    3  times  4 


"  First  we  took  from  the  93,  20  times  4,  and  had  13  left. 
"  Then  we  took  from  the  13,  3  times  4,  and  had  1  left. 
"  Altogether  we  took  from  the  93,  23  times  4,  and  had 
1  left." 

A  little  work  of  this  kind  will  suffice  to  justify  the  process 
to  the  brighter  children;  a  few  will  grasp  it  fully.  The  dullards 
will  not  understand  it  all;  they  want  the  clear-cut  rule,  and 
explanations  merely  worry  them. 

Now  consider  an  ordinary  long  division  sum;  say,  divide 
45329  by  87.  Let  the  children  write  out  the  87  times  table, 
to  9  X  87. 


1  X  87  =  87 

2  x  87  =  174 

3  x  87  =  261 


4  x  87  =  348 

5  X  87  =  435 

6  X  87  =  522 


7  X  87  =  609 

8  X  87  =  696 

9  X  87  =  783 


(In  making  a  table  like  this  note  that  3  times  =  2  times 


40  CRAFTSMANSHIP  IN  MATHEMATICS 

+  1  time,  5  times  =  3  times  +  2  times,  &c.,  and  so  save 
the  labour  of  multiplying;  only  multiplication  by  2  is  necessary; 
all  the  rest  is  easy  addition.) 


87)45329 

"  87's  into  4?  won't  go:   4  is  not  big  enough;  put  a  dot 

over  it. 
"  87's  into  45?    won't  go:    45  is  not  big  enough;   put  a 

dot  over  the  5. 
"  87's  into  453?    will  go,  because  453  is  bigger  than  87. 

How  many  times?" 

Look  at  the  table,  and  take  the  biggest  number  (435)  that 
can  be  subtracted  from  453.  The  435  is  5  times  87.  Place 
the  5  over  the  3  in  the  dividend,  write  the  435  under  the  453, 
and  subtract;  the  difference  is  18. 

Bring  down  the  2  from  the  dividend,  placing  it  to  the  right 
of  the  18,  making  182.  Look  at  the  table  again,  and  take  the 
biggest  number  (174)  that  can  be  taken  from  the  182,  &c. 

"521 

87)45329 

435!  : 

~-T82; 

174: 

"^9 

87 


"  Thus  we  know  that  87  is  contained  521  times  in  45329, 
and  that  there  is  2  (the  Remainder)  to  spare. 

"  What  is  the  biggest  R  we  could  have?  Could  it  be  87? 
Why  not?" 

Teach  the  children  the  usual  verification  check:  multiply 
the  divisor  by  the  quotient,  add  R  to  the  product,  and  so 
obtain  the  original  dividend. 

(Do  not  forget,  when  introducing  formulae  later,  to  utilize 
the  D,  d,  Q,  and  R.  D  =  dQ  or  dQ  +  R.) 


ARITHMETIC:    FIRST  FOUR  RULES  41 

Now  pour  a  little  gentle  scorn  upon  making  out  a  special 
multiplication  table  for  every  division  sum:  "  We  must 
give  up  such  baby  tricks  ".  But  that  leads  us  to  what  some 
beginners  in  division  find  very  difficult — how  to  tell  the 
number  of  times  a  big  divisor  will  go  into  one  of  the  numbers 
derived  from  the  dividend: 

69)342 

"  Instead  of  saying  *  69's  into  342  ',  we  cut  off  the  last  figure 
of  the  69  and  of  the  342  and  say  6's  into  34  instead.  This 
seems  to  be  5,  but  the  5  may  be  too  big,  because  of  the  carry 
figure;  we  find  it  is  too  big,  so  we  try  4  instead. " 

Warn  the  children  that  if,  after  subtracting  at  any  step, 
they  have  a  difference  greater  than  the  divisor,  the  figure 
they  have  just  put  into  the  quotient  is  too  small.  Rub  this 
well  into  the  dullards. 

Warn  them,  too,  that,  above  every  figure  in  the  dividend, 
they  must  place  either  a  dot  or  a  figure  for  the  quotient. 

"•341,  R  =  267. 
329)112466 

Similarly  in  short  division  except  that  the  dots  and  figures 
go  below: 

7)13259 

•1894,  R  =  1. 


A  dot  is  not  a  very  suitable  mark,  owing  to  confusion  with  a 
decimal  point;  it  is,  however,  in  common  use.  /  If  no  mark  is 
used,  figures  get  misplacedand  errors  made.  The  marks 
may  be~  dropped  later.  ~~~ 

The  advantage  of  the  method  of  placing  the  quotient 
over  jnstead  of  totheright  of  the  dividend,  jsjliat  children 

write  down  noughts  when  these 


are  reguired^.  ' 

Eet  division  by  factors  stand  over  until  the  senior  school. 
The  calculation  of  the  remainder  is  puzzling  to  beginners. 
Divide  34725  by  168.  Suitable  factors  of  168  :  4  X  6  x  7. 


42  CRAFTSMANSHIP  IN  MATHEMATICS 

434725  units. 

8681  fours,  R  =  1  unit. 


u    OUOA  iuurs,  r\  =  JL  unit. 

7  '  1446  twenty-fours,  R  =  5  fours. 
• '  206  one  hundred  and  sixty-eights,  R  =  4  twenty-fours. 

Total  Remainder  =  (24  x  4)  +  (4  X  5)  +  1 

=  117. 
Quotient         =  206,  R  =*  117. 

Avoid  the  Italian  method,  except  perhaps  with  A  Sets. 
With  average  children  the  method  is  productive  of  great 
inaccuracy. 

In  fact,  avoid  all  short  cuts  until  main  rules  are  thoroughly 
mastered.  For  instance  if  a  boy  has  to  multiply  by  357, 
do  not  teach  him  to  multiply  by  7,  and  then  multiply  this 
first  partial  product  by  50  to  obtain  his  second  partial  product; 
it  is  simply  asking  for  trouble. 

Of  course,  practised  mathematicians  do  these  things,  but 
we  have  to  think  of  beginners.  Teach  a  straightforward 
method,  and  stick  to  it.  Hints  as  to  "  neat  dodges  "  and 
about  "  short  cuts  "  are  for  the  few,  not  for  all. 


CHAPTER  V 

Arithmetic :    Money 

Money  Tables 

No  part  of  arithmetic  is  more  important  than  the  various 
manipulative  processes  of  money.  It  is  with  us  every  day 
of  our  lives,  and  accuracy  is  indispensable.  The  ordinary 
money  tables  must  be  known,  and  thus  more  ding-dong  work 
is  necessary.  This  is  mainly  a  question  of  a  knowledge  of 
the  12  times  table.  Five  minutes'  brisk  mental  work  twice 


ARITHMETIC:  MONEY  43 

a  day  will  pay,  sometimes  with  and  sometimes  without  the 
blackboard,  and  sometimes  on  paper. 
Associate  with  the  12  times  table: 


1  x  12  =  12 

2  X  12  =  24 

3  X  12  =  36 


I2d.  =  Is. 
24<*.  =  2*. 
36 d.  =  3s. ,  &c.  &c. 


Day  by  day  drill: 

SQd.  =  ?       83<f.  =  ?       S4d.  =  ?       89<f.  =  ?     &c. 
Pence  in  7s.  ?        in  7s.  3d.  ?        in  9s.  9J.  ?      &c. 

and  so  every  day  until  accurate  answers  up  to  144*/.  are 
instantaneous.  If  the  boys  are  familiar  with  two  definite  land- 
marks in  each  "  decade  ",  20  and  24,  30  and  36,  40  and  48, 
&c.,  the"  necessary  additions  for  the  other  numbers  of  each 
decade  are  simple. 

Associate  the  farthings  table  with  the  4  times  table,  and 
the  shillings  table  with  the  20  times  table,  which  is  easily 
learnt. 

Let  every  mental  arithmetic  lesson  at  this  stage  include 
simple  addition  and  subtraction  of  money,  especially  the 
addition  of  short  columns  of  pence. 

Elementary  facts  concerning  the  coinage  should  be 
associated  with  the  money  tables,  and  in  this  connexion  do 
not  forget  guineas  (which  often  figure  in  subscriptions  and 
in  professional  fees)  and  Bank  of  England  notes. 

At  a  later  stage  the  boys  should  be  taught  such  common- 
place facts  about  the  coinage  as  every  intelligent  person 
ought  to  know,  e.g.  the  nature  of  "  standard  "  gold  and 
silver,  degrees  of  "  fineness  ",  the  nature  of  the  present 
legalized  alloy  for  "  silver  "  coinage,  the  nqyket  prices  of 
pure  gold  and  silver,  the  nature  of  bronze;  alloys;  tokens; 
the  Mint. 

Reduction 

Reduction  is  not  likely  to  give  serious  trouble,  if  the 
tables  are  known.  The  commonest  mistake  is  to  multiply 


44  CRAFTSMANSHIP  IN  MATHEMATICS 

instead  of  divide,  or  vice  versa.  Teach  the  boy  to  ask  himself 
every  time  whether  the  answer  is  to  be  smaller  or  larger\ 
if  smaller,  to  divide;  if  larger,  to  multiply.  But  "  guineas  to 
pounds  ",  and  the  like,  is  a  type  of  sum  that  baffles  the  slow 
boy  and  requires  special  handling. 

Subtraction 

There  is  something  to  be  said  for  avoiding,  at  first 
writing  farthings  in  the  usual  fractional  form  and  for  giving 
them  a  separate  column: 

£  *•  d.  f. 

47  14  6     1 

21  19  4     3 

25  15  1   ~2 


let  the  children  use  the  fractional  forms  a  little  later,  when 
they  may  be  made  a  useful  introduction  to  fractions.  The 
alternative  is  to  omit  farthings  altogether  in  the  early  stages. 

Multiplication 

How  is  this  to  be  done?  Whatever  method  is  adopted, 
a  percentage  of  inaccurate  answers  seems  almost  to  be  in- 
evitable. We  set  out  a  sum  by  each  of  the  4  methods  commonly 
used.  Multiply  £7,  15$.  10W.  by  562. 

I.  £    s.     d. 

7     15     10J 

10 

77     18       9    =  £7     15     10J  X  10 

10 

779      7       6    =  £7     15     10J  X  100 
5 


3896  17  6  =  £7  15  10J  x  500 

15  11  9  =  £7  15  10J  x       2 

467  12  6  =  £7  15  10J  X     60 

4380  1  9  =  £7  16  10J  X  562 


ARITHMETIC:  MONEY 


II. 


662 
16 


20|  84305. 

£421,105.  Od. 


12[6620</. 

20   468^. 


2562/z. 


£23,  85.  4d. 


1228U. 


235.  5d. 


45 


£7 

0 

0 

X 

662  = 

£3934 

0 

0 

£0 

16 

0 

X 

662  = 

£  421 

10 

0 

£0 

0 

10 

X 

562  = 

£  23 

8 

4 

£0 

0 

t 

X 

562  = 

£   i 

3 

5 

£7 

16 

10* 

X 

562  = 

£4380 

1 

9 

III. 


562  at  £7,  155. 

7 

lojd. 

3934 

0 

0  = 

£7 

0 

0 

X 

562 

* 

of£l 

281 

0 

0  = 

0 

10 

0 

X 

562 

t 

of  10/- 

140 

10 

0  = 

0 

5 

0 

X 

562 

¥ 

of  5/- 

17 

11 

3  =r 

0 

0 

7J 

X 

562 

i 

tv 

of  5/- 

7 

0 

6  = 

0 

0 

3 

X 

562 

£4380 

1 

9  = 

£7 

15 

iot 

X 

562 

IV. 


£7,  165.  10H  x  562 
=  £7-79375  x  562 
=  £4380-0875 
=  £4380,  15.  9d. 


My  own  experience,  and  this  corresponds  to  the  results  of 
many  inquiries,  is  that  the  second  method  produces  the  best 
results;  then  the  third  ("  practice  ")  method,  provided  that 
pupils  have  been  well  drilled  in  aliquot  parts  (though  some 
always  seem  to  find  division  more  difficult  than  multiplication); 
then  the  first  method.  The  last  is  a  good  method  for  older 
pupils  who  have  learnt  to  decimalise  money  readily,  but 
not  for  younger  pupils  or  for  slower  older  pupils. 

The  ordinary  method  (the  first  method)  is  curiously 
productive  of  errors;  in  the  course  of  a  long  experience  I 
have  never  known  a  whole  class,  without  exception,  get  a 
sum  right  by  this  method,  even  after  they  had  had  several 
months'  practice.  The  second  method  generally  leads  to 
untidy  and  unsystematic  marginal  work.  This  marginal 


46  CRAFTSMANSHIP   IN  MATHEMATICS 

work  should  be  made  an  integral  part  of  the  working  of  the 
sum,  and  should  not  be  looked  upon  as  scrap  work. 


Division 

Dr.  Nunn's  suggestion  that  the  process  of  working  may 
be  set  out  in  the  following  way  may  well  be  followed.  All 
pounds  are  kept  in  one  vertical  column,  shillings  in  another, 
and  so  on.  It  is  very  neat  and  compact.  Allow  plenty  of 
space  across  the  paper.  Example:  Divide  £3541,  14s. 
by  47. 

£  s.  d.  f. 

••75  7  1  1 


47)3541 
329 

14        9        2 
->320      (->60     ,-»88 

•251 

334 

69 

90 

235 

329 

47 

47 

•16  X  20- 

••5  X  12- 

22  X  4- 

43  R 

Answer:  £75,  7s.  \^d.  and  43  farthings  over. 

Other  methods  have  been  devised,  but  this  old  method  is 
probably  best  and  most  readily  learnt. 

To  ensure  a  full  understanding  of  the  nature  of  the 
"  remainder  "  a  sum  like  the  above  should  be  followed  up 
by  two  others: 

1.  Take  43  farthings  from  the  dividend;   then  divide 

again. 

2.  Add  4  farthings  to  the  dividend;  then  divide  again. 

Even  slower  boys  can  usually  explain  the  (to  them)  rather 
surprising  new  answers. 


WEIGHTS  AND  MEASURES  47 

CHAPTER   VI 

Weights  and  Measures 

Units  and  Standards 

Consider  what  weights  and  measures  are  used,  to  what 
extent,  and  how,  in  practical  life.  Teach  these,  and  teach 
them  well,  and  let  all  the  rest  go.  A  coal  merchant  concerns 
himself  with  tons,  cwt.,  qr.,  never  with  Ib.  and  oz.;  a  grocer 
with  cwt.,  qr.,  Ib.,  oz.,  never  with  drams  and  rarely  with  tons; 
a  farmer  with  acres,  quarter-acres,  and  perches;  a  builder 
with  yards,  feet,  and  inches;  a  surveyor  with  chains  and  links; 
and  so  generally.  The  completer  tables  of  weights  and 
measures  are  generally  given  in  the  textbooks  as  a  matter 
of  convenience,  but,  in  practical  life,  the  whole  table  is 
seldom  wanted  by  any  one  person.  A  teacher  who  gives  boys 
reduction,  multiplication,  or  division  sums,  say,  from  tons 
to  drams  (or  even  to  ounces),  or  from  square  miles  to  square 
inches,  is  simply  proclaiming  aloud  his  incompetence:  perhaps 
he  is  the  slave  of  some  stupidly  written  textbook;  certainly 
he  is  lacking  in  judgment.  The  main  thing  is  to  make  the 
boys  thoroughly  familiar  with  the  few  weights  and  measures 
that  are  commonly  used,  and  to  give  them  a  fair  amount  of 
practice  in  the  simpler  transformations  of  comparatively  small 
quantities;  and  to  let  all  the  rest  go. 

Teach  clear  notions  of  units  and  standards.  Show  how 
unintelligent  we  British  people  have  always  been  in  our 
choice  of  units.  We  have,  for  instance,  determined  our  inch 
by  placing  three  grains  of  barley  in  line;  we  have  selected 
our  foot,  because  12  of  the  inches  roughly  represent  the 
length  of  a  man's  foot;  we  have  determined  our  smallest 
weight  (the  grain),  by  adopting  the  weight  of  a  dried  grain 
of  wheat.  That  such  things  vary  enormously  did  not  trouble 
our  forefathers  at  all.  Tell  the  boys  that  at  one  time  the 
French  people  had  similar  unsatisfactory  weights  and  measures 


48  CRAFTSMANSHIP  IN  MATHEMATICS 

but  that  now  they  have  changed  to  a  system  much  more 
rational. 

Let  the  various  tables  be  learnt  and  learnt  perfectly. 


Weight 

1.  Avoirdupois  (not  used  for  the  precious  metals). — Let 
the  table  to  be  learnt  include  the  oz.,  Ib.,  qr.,  cwt.,  ton. 
Note  that  the  standard  weight  is  the  pound,  which  consists 
of  7000  grains.  (A  dried  grain  of  wheat,  though  roughly  a 
grain  in  weight,  is  not,  of  course,  a  standard.  A  grain  is 
1/7000  part  of  a  pound.) 

Teach  the  stone  as  a  separate  item:  normally  14  lb.,  but  for 
dead  meat,  8  Ib. 

Give  easy  sums  for  practice  in: 

(1)  tons,  cwt.y  qr.  (coal  and  heavy  goods). 

(2)  cwt.,  qr.,  Ib.  (wholesale  grocery). 

(3)  cwt.,  st.,  Ib.  (wholesale  meat  purchases). 

(4)  Ib.,  oz.  (retail  grocery  and  meat). 

Note  that  an  oz.  of  water  or  any  other  fluid  is  an  avoirdupois 
ounce,  like  the  ounce  of  any  common  solid,  and  contains 
7000/16  or  437£  grains. 

In  making  up  arithmetical  examples,  utilize  as  far  as 
possible  the  quantities  (sacks,  bags,  chests,  &c.)  representing 
the  unit  purchases  of  tradesmen  and  others,  though  the 
problem  given  will  often  depend  on  the  locality.  For  instance, 
problems  on  crans  and  lasts  of  herrings,  or  on  trusses  of  hay 
or  straw,  would  be  quite  inappropriate  in  big  inland  towns. 
The  teacher  should,  for  problem  purposes,  make  a  note  of 
points  like  the  following:  weight  of  a  chest  of  tea,  f  cwt.; 
sack  of  coal  or  of  potatoes,  1  cwt.;  bag  of  flour,  1J  cwt.; 
bag  of  rice,  1£  cwt.;  truss  of  straw,  36  Ib.;  truss  of  new  hay, 
60  Ib.,  of  old  hay  56  Ib.;  a  brick,  7  Ib.;  1000  bricks,  3£  tons; 
100  Ib.  of  wheat  produces  70  Ib.  of  flour  which  produces 
91  Ib.  of  bread;  and  so  on.  Everyday  quantities  of  this  kind 


WEIGHTS  AND  MEASURES  49 

give  a  reality  to  problems  in  arithmetic  that  even  the  non- 
mathematical  boy  appreciates. 

2.  Troy,  used   by  jewellers. — Let  the  table  to  be  learnt 
include  the  grain,  dwt.,  oz.,  Ib.    It  is  important  to  remember 
that  the  Troy  ounce  is  heavier  than  the  common  (avoirdupois) 
ounce,  since  it  contains  480  grains,  as  against  437^.     But 
the  Troy  pound  is  lighter  than  the  common  pound,  since  it 
contains  only  12  ounces  and  therefore  5760  grains,  as  against 
7000  in  the  avoirdupois  Ib. 

3.  Apothecaries'. — The  old  weights  have  gone  out  of  use. 
Drugs  are  generally  used  in  very  small  quantities,  and  the 
basic  weight  is  the  grain  (the  grain  is  a  constant  weight  for 
all  purposes).     A  quantity  of  drugs  weighing  more  than  a 
few  grains  is  expressed  as  a  fraction  of  an  ounce  avoirdupois. 

N.B. — Ignore  the  avoirdupois  dram  (^  oz.)  and  the 
druggists'  old  scruple  and  drachm  weights.  The  dram  and 
drachm  were  not  the  same. 


Length 

Let  the  main  table  to  be  learnt  include  the  in.,  ft.,  yd., 
pole,  fur .,  mile,  and  let  the  link,  chain,  and  fur.  be  included  in 
a  separate  table.  Remind  the  boys  that  the  chain  is  the 
length  of  a  cricket  pitch. 

Give  easy  sums  for  practice  in: 

(1)  yd.,  ft.,  in.  (builders,  &c.). 

(2)  poles,  yd.,  ft.  (farmers,  &c.). 

(3)  miles,  chains,  links  (surveyors). 

(4)  miles,  yd.  (road  distances,  &c.). 

Measures  that  may  be  drawn  from  practical  life  for 
problem  use  are  almost  innumerable.  The  sizes  of  battens, 
deals,  and  planks  will  be  learnt  in  the  manual  room;  notes 
of  the  sizes  of  other  materials  used  by  builders — slates, 
glass,  door-frames,  &c.,  &c. — may  be  made  from  time  to  time; 
size  of  an  ordinary  brick,  8f "  X  4£"  X  2£"  (note  the  \"  all 

(E291)  5 


60  CRAFTSMANSHIP  IN  MATHEMATICS 

round  for  jointing),  square  tile,  9f"  X  9f  "  X  1"  or  6"  X  6" 
X  1";  machine-printed  wall-paper,  11  \  yd.  X  21";  hand- 
printed, 12  yd.  X  21";  French,  9  yd.  X  18";  sheets  of 
paper,  foolscap,  17"  X  13|"  (see  Whilaker  for  other  sizes); 
bound  books,  foolscap  8vo,  6f"  X  4|"  (see  Whitaker)\ 
skein  of  yarn  =  120  yd.,  hank  =  7  skeins;  railway  gauge 
4'  8|"  (12'  of  roadway  for  single  track,  23'  for  double);  equator, 
24,902  miles;  polar  diameter,  7926  miles;  fathom,  6';  knot, 
6080'  (40  knots  ==  46  miles).  These  are  only  a  tithe  of  the 
everyday  measurements  that  may  be  used  for  making  up 
problems.  Such  problems  are  far  more  valuable  than  the 
hackneyed  reduction  sums  given  in  the  older  textbooks. 

N.B. — The  ell,  league,  and  such  foreign  lengths  as  the 
verst,  may  be  ignored.  The  cubit  is  worth  mentioning. 

Area 

Let  the  table  to  be  learnt  include  the  sq.  in.,  ft.,,  yd.  pole, 
the  rood,  the  acre,  sq.  mile.  It  is  useful  to  remember  that  an 
acre  =  10  sq.  chains,  or  a  piece  of  ground  220  yd.  X  22  yd., 
or  a  piece  just  about  70  yd.  square. 

Give  easy  sums  for  practice  in: 

(1)  sq.  miles,  acres  (areas  of  counties,  &c.). 

(2)  ac.,  ro.,  sq.  poles  (farmers,  &c.). 

(3)  sq.  yd.,  sq.ft.,  sq.  in.  (builders,  &c.). 

Familiar  areas  for  problem  purposes:  Lawn  tennis 
court,  78'  X  36'  or  78'  X  27';  Association  football  ground, 
120  yd.  X  80  yd.;  Rugby,  110  yd.  X  75  yd.;  croquet  lawn, 
105'  X  84';  Badminton  court,  44'  X  20';  &c. 


Volume 

Table:  c.  in.,  ft.,  yd. 

Let  sums  for  practice  be  of  the  simplest,  e.g.  the  number 
of  cubic  yards  of  earth  excavated  from  a  trench;  the  number  of 


WEIGHTS  AND  MEASURES  51 

cubic  feet  of  brickwork  in  a  wall;  the  cubic  capacity  of  a 
room  or  of  a  building;  the  volume  of  the  Earth  in. cubic 
miles. 

Capacity 

1.  Liquids. — Table:     gill,   pt.,    qt.,   gall.     Casks    have    a 
variety  of  names:    barrel  of  ale  —  36  gall.;  hogshead  of  ale 
—  54  gall.,  of  wine  =  63  gall.,  &c.    A  wine  bottle  =  £  gall.; 
Winchester  quart  =  \  gall. 

2.  Dry  Goods  (corn,  &c.). — Table:   peck,  bushel,  quarter. 
The  quarter-peck  is  called  a   "  quartern  ";    the  half-peck  is 
the  equivalent  of  a  gallon.  The  gallon  is  a  kind  of  link  between 
the  liquid  and  dry  measures. 

There  is  now  a  tendency  to  substitute  weight  for  measure. 
Problems  on  capacity  reduction  are  hardly  worth  doing, 
except  small  problems  that  may  be  done  mentally.  But 
problems  involving  transformations  between  capacity  and 
weight  are  common,  and  ample  practice  is  necessary.  N.B. — 
1  gall,  of  water  weighs  10  Ib.  "  A  pint  of  pure  water  weighs 
a  pound  and  a  quarter." 

Liquid  medicine  measure  (mainly  solutions  in  water). 

Table:  60  minims  =  1  fluid  drachm. 

8  fluid  drachms  =  1  fluid  ounce. 
20  fluid  ounces  =  1  pint. 

The  fluid  ounce  is  the  same  as  the  common  (avoirdupois) 
ounce,  and  therefore  weighs  437^  grains.  But  it  contains 
480  minims,  and  therefore  a  minim  weighs  rather  less  than 
a  grain.  The  minim  may  be  thought  of  as  a  "  drop  ",  though 
of  course  drops  vary  greatly  in  size. 

Doctors'  prescriptions  may  be  discussed,  rather  than 
sums  worked.  If  a  solid  drug  is  prescribed,  the  amount  is 
expressed  in  grains  or  in  fractions  of  an  ounce;  if  liquid,  then 
minims,  drachms,  or  ounces. 


52  CRAFTSMANSHIP  IN  MATHEMATICS 

Time 

The  second,  minute,  hour,  day,  week,  give  little  trouble. 
The  variable  month  requires  careful  explanation.  Teach 
the  doggerel  "  Thirty  days  hath  September  ",  &c.,  or  furnish 
some  alternative  mnemonic.  Explain  "  leap  "  year  and  its 
determination. 

Few  problems  of  reduction  are  necessary,  and  these  should 
be  easy.  A  few  on  the  calendar  are  advisable,  and  a  few 
dealing  with  speeds. 

Useful  Memoranda 

Other  useful  memoranda  for  problem-making. — (The  quan- 
tities are  approximations  only  and  should  be  memorized. 
They  are  useful  when  closer  approximations  have  to  be 
evaluated): 

1  cubic  foot  of  water  —  6J  gall.  —  62  J  Ib. 
1  cubic  inch  of  water  =  252  £  grains. 

A  common  cistern  4'  x  3'  X  2* '  =  30 c.  ft.  —  187  gall.  =  £  ton. 
1  ton  of  water  =  36  c.  ft.  =  224  gall. 
1  gallon  of  water  =  277|  c.  in.  =  10  Ib. 

1  ton  of  coal  occupies  about  40  c.  ft.  (hence  25  tons  need  a 
space  10'  X  10'  X  10'). 

Wall    Charts 

A  few  permanent  charts  are  useful  on  a  wall  of  the  class- 
room where  weights  and  measures  are  taught:  an  outline 
plan  of  (1)  the  town  or  village  showing  the  over-all  dimen- 
sions, length,  breadth,  and  area;  (2)  the  school-site  and 
buildings;  (3)  the  school  itself;  (4)  the  actual  classroom. 

(5)  Diagram    to    scale    to    show    that    5-|    yd.  X  5|-  yd.  = 
30  J   sq.  yd.    (often    used    for  a    first    lesson    in   fractions). 

(6)  A  chart  giving  the  weights  of  a  few  familiar  objects  in 
and  about  the  school,  and  the  capacities  of  a  few  others.    See 
that  these  charts  are  used  and  known. 


WEIGHTS  AND  MEASURES  53 


The  Metric  System 

Some  knowledge  of  this  system  is  necessary,  if  only 
because  of  the  work  in  the  physical  laboratory.  The  be- 
ginner may  be  shown  a  metre  measure  side  by  side  with  a 
yard  measure,  and  simply  be  told  that  it  is  rather  longer, 
and  had  its  origin  in  France.  As  the  boy  goes  up  the  school 
he  will  learn  that  its  length  is  about  39-37  in.,  and  is  the 
measured  fraction  of  a  quadrant  of  the  earth's  surface.  A 
little  later  still,  he  will  be  told  how  the  French  measured 
the  actual  length  of  an  arc  of  one  of  their  meridians,  and 
how  they  determined  the  latitude  of  each  place  at  the  end  of 
the  arc.  This  easy  astronomical  problem  is  usually  worked 
out  in  a  Fifth  Form  geography  lesson. 

The  cubic  decimetre  and  the  litre,  the  cubic  centimetre 
and  the  gram,  are,  as  derivations  of  the  initial  metre,  always 
a  source  of  interest  to  boys. 

The  boys  should  memorize  the  few  usual  approximate 
equivalents  between  the  British  and  metric  systems,  e.g. 
1  metre  =  39-37  in.;  1  kilogram  =  2-2  lb.;  1  litre  =  1-76 
pints;  1  gram  =  15-43  grains;  1  are  =  -£$  acre.  With  these 
they  can  quickly  estimate  quantities  in  terms  of  metric  units. 
For  instance,  a  Winchester  quart  will  hold  4/1-76  litres 
=  2-27  litres  =  2270  c.  c.;  1  hectare  =  2£  acres;  and  so  on. 

But  do  not  forget  to  enter  a  defence  in  favour  of  our  own 
system  of  weights  and  measures,  if  the  metric  system  is 
advocated  on  purely  scientific  grounds.  Sixth  Form  boys 
are  always  interested  in  this.  In  the  first  place,  the  metre 
was  not  measured  accurately;  in  the  second  place,  it  is  a 
local  and  not  a  universal  unit;  it  depends  upon  the  length 
of  a  particular  meridian  in  a  particular  country.  The  meridian 
the  French  measured  was  an  ellipse,  not  a  circle,  and  not 
a  true  ellipse  at  that.  Had  they  utilized  the  polar  axis  (a 
fixed  length)  instead  of  a  meridian  (a  variable  length),  their 
unit  would  have  been  more  scientific,  for  it  would  have  been 
universal,  and  it  could  have  been  measured  more  accurately. 


54  CRAFTSMANSHIP  IN  MATHEMATICS 

The  length  of  the  polar  axis  is  very  nearly  500,500,000  in., 
so  that  the  inch  already  bears  a  simpler  relation  to  the  polar 
axis  than  the  metre  does  to  its  own  meridian  quadrant.  If  we 
adopted  a  new  inch,  viz.  1/500,000,000  of  the  polar  axis, 
it  would  make  but  a  very  slight  change  in  our  linear  measure- 
ments, and  then,  curiously  enough,  a  cubic  foot  of  water 
would  weigh  almost  exactly  1000  oz.  (instead  of  997).  Our 
present  ounce  weight  would  have  to  be  increased  by  only 
Y§-  part  of  a  grain!  Moreover,  the  new  cubic  foot  would 
contain  exactly  100  half-pints.  Such  a  new  system  would 
be  incomparably  more  scientific  than  the  metric  system. 

Thus  the  opponents  to  the  adoption  of  the  metric  system 
have  sound  arguments  to  support  their  views.  The  metre 
has  on  its  side  the  virtue  of  being  the  basic  unit  of  a  con- 
venient and  simple  system;  but  scientifically  it  is  a  poor 
thing. 

There  is  no  need  for  the  boys  to  learn  the  metric  tables. 
But  they  should  learn  the  three  Latin  prefixes  deci,  centi,  milli, 
and  know  that  these  represent  fractions;  and  the  three  Greek 
prefixes  deca,  hecto,  kilo  representing  multiples.  These  learnt 
thoroughly,  the  tables  as  such  are  unnecessary.  But  with 
three  or  four  exceptions  the  multiples  and  sub-multiples  are 
hardly  ever  wanted. 

The  best  exercises  on  the  metric  system  are  those  based 
on  laboratory  operations. 


CHAPTER  VII 

Factors  and  Multiples 

The  term  "  factor  "  and  "  multiple  "  should  be  used 
when  the  tables  are  being  taught,  though  without  formal 
definition.  "  3  X  7  =  21;  we  call  3  and  7  factors  of  21." 

Give  me  a  factor  of  6?  3;   another?  2;   a  factor  of  30?  2; 


FACTORS  AND  MULTIPLES  55 

another?  3;  another?  5,  A  multitude  of  people  means 
many  people,  and  a  multiple  of  a  number  means  a  bigger 
number  containing  it  many  times,  though  "  many  "  may  not 
be  greater  than  2.  Now  think  of  your  5  times  table.  Give 
me  a  multiple  of  5?  15;  another?  30;  another?  35. — After  a 
little  of  this  work,  the  terms  factor  and  multiple  will  become 
part  of  the  boys'  familiar  vocabulary.  "  Common  "  factor 
and  multiple  will  come  later.  One  idea  at  a  time. 


Tests  of  Divisibility 

Prime  Factors. — Tests  of  divisibility  for  2,  3,  5,  and  10^ 
may  readily  be  given  in  the  Junior  School  or  Department; 
those  for  4,  8,  9,  11,  12,  25,  125,  a  year  or  two  later.  At  first, 
give  the  rules  dogmatically. 

"  A  number  is  divisible  by  2  if  it  is  an  even  number. 

„          „  „  3  if  the  sum  of  its  digits  is 

divisible  by  3. 

„          „          „  5  if  it  ends  in  a  5 

„          „          „  10  if  it  ends  in  a  0." 

Justification,  not  "  proofs  ",  of  such  rules  is  commonly 
given  in  Form  IV.  The  reasoning,  which  is  quite  simple, 
depends  on  the  principle  that  a  common  factor  of  two 
numbers  is  a  factor  of  their  sum  or  their  difference.  Never 
mind  the  general  proof;  at  this  stage  merely  justify  the 
principle  by  considering  a  few  particular  instances,  and 
these  readily  emerge  from  the  multiplication  table;  for 
instance: 


5  fours  =  20 
7  fours  =  28 


12  fours  =  48 


9  fives  =  45 
7  fives  =  35 
2  fives  =  K) 


We  know  that  5  fours  added  to  7  fours  make  12  fours,  i.e. 

4  is  a  factor  of  20  and  of  28,  and  is  also  a  factor  of  20  +  28. 

Again,  we  know  that  7  fives  from  9  fives  is  2  fives,  i.e. 


66  CRAFTSMANSHIP  IN  MATHEMATICS 

5  is  a  common  factor  of  35  and  45,  and  is  also  a  factor  of 
45  -  35. 

Divisibility  by  2. — Consider  any  even  number,  say  754; 
754  =  750  +  4.  Since  2  is  a  factor  of  10  and  therefore  of  the 
multiple  750,  and  is  also  a  factor  of  4,  it  is,  by  our  rule,  a 
factor  of  750  +  4  or  754. 

Divisibility  by  5. — Consider  any  number  ending  in  5, 
say  295;  295  =  290  +  5.  Since  5  is  a  factor  of  10  and  there- 
fore of  the  multiple  290,  and  is  also  a  factor  of  5,  it  is,  by 
our  rule,  a  factor  of  290  +  5  or  295. 

Divisibility  by  3. — Consider  any  number,  say  741. 

741  =  700  -}-  40+1 

=  (100  x  7)  +  (10  X  4)  +  1 
=  (99  X  7)  +  7  +  (9  X  4)  +  4  +  1 
=  (99  X  7)  +  (9  X  4)  -{-  7  -f  4  +  1 
=  (99  X  7)  +  (9  X  4)  +  12. 

Now  3  is  a  factor  of  9  and  therefore  of  all  multiples  of  9; 
it  is  also  a  factor  of  12.  Since  3  is  a  factor  of  99  X  7  and  of 
9x4  and  of  12,  it  is  a  factor  of  their  sum,  i.e.  of  741. 
Hence,  &c. 

A  formal  proof  of  the  principle  used  should  be  associated 
with  the  algebra  later. 

The  justification  of  the  rule  for  4  and  25,  8  and  125, 
and  9  is  equally  readily  understood,  but  that  for  11  is  a 
little  more  difficult. 

Primes  and  Composite  Numbers 

Quite  young  boys  quickly  see  the  distinction  between  a 
prime  and  a  composite  number  and  are  always  interested 
in  the  sieve  of  Eratosthenes. 

Third  Form  boys  should  be  made  to  memorize  the  squares 
of  all  numbers  up  to  20;  132  =  169;  172  =  289;  &c.  (The 
squares  of  13,  17,  and  19  must  really  be  learnt;  142,  162, 
and  182  can  be  mentally  calculated  in  a  second  or  two,  if 
forgotten.)  Then  give  a  little  mental  practice  in  extracting 


FACTORS  AND  MULTIPLES  57 

square  roots:  of  81?  of  256?  of  361?  (Mention  that  the  root 
sign  (vO  we  use  is  merely  a  badly  written  form  of  the  initial 
letter  R.) 

Make  the  class  write  down,  in  order,  the  successive  pairs 
of  factors  of,  say,  36: 

2  X  18 

3  x  12 
4x9 
6X6 

I  9  X  4 
112  x  3 
\18  X  2 

Then  point  out  that  the  second  column  is  the  first  column 
reversed,  and  that  the  3  lower  horizontal  lines  are  the  3 
upper  horizontal  lines  reversed.  Hence  when  we  have  to 
write  dowu  the  factors  of  36,  we  need  not  proceed  beyond 
the  fourth  line,  viz.  6x6,  for  then  we  already  have  all  the 
factors;  and  the  6,  the  last  trial  number,  is  \/36.  The  boys 
can  now  appreciate  the  common  rule:  When  resolving  a 
number  into  factors,  it  is  unnecessary  to  carry  our  trials  beyond 
its  square  root,  unless  the  number  is  not  a  perfect  square, 
and  then  it  is  advisable  to  consider  the  next  square  number 
beyond  it. 

For  instance,  write  down  all  the  factors  of  120;  120  is 
not  a  square  number,  but  the  next  square  number  is  121, 
the  square  root  of  which  is  11.  Hence  we  need  not  proceed 
with  our  trial  numbers  beyond  11,  but  as  11  does  not  happen 
to  be  a  factor,  we  do  not  proceed  beyond  10.  Thus  by  trial 
we  find  that  2,  3,  4,  5,  6,  8,  and  10  are  factors;  and,  dividing 
120  by  each  of  these,  we  obtain  other  factors  which  pair  off 
with  them.  The  14  factors  of  120  are 

2      3      4      5      6      8    10 
60    40     30     24     20     15     12 

The  next  step  is  to  teach  factor  resolution  by  trials  of 
prime  numbers  only.  This  causes  no  additional  difficulty, 
but  the  boys  should  recognize  at  once  all  the  prime  numbers 


68  CRAFTSMANSHIP  IN  MATHEMATICS 

up  to,  say,  41  (1,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31, 
37,  41). 

"  Write  down  the  factors  of  391.  The  next  square  number 
beyond  is  400.  Hence  we  need  not  carry  our  trials  beyond 
-V/400  or  20.  By  trial  we  find  that  the  lowest  factor  is  17. 
By  division  we  obtain  23,  which  we  recognize  as  another 
prime  number.  Thus  17  and  23  are  the  only  factors. " 

I  have  known  a  C  Set  of  a  Fourth  Form  become  expert  in 
factorizing  3-figure  numbers,  after  one  lesson.  It  is  work 
that  most  boys  like. 

"  Express  360  as  a  product  of  factors  which  are  all  prime. 
— Divide  by  the  lowest  prime  number,  2  if  possible;  and 
again  and  again  if  necessary;  then  by  the  next  prime,  3, 
if  possible;  and  again  and  again,  if  necessary.  Then  by  5, 
if  possible;  then  7;  then  11;  and  so  on. 

360  =  2  x  180 

=  2  X  2  X  90 

=  2x2x2x45 

=  2x2x2x3x15 

=  2x2x2x3x3x5. 

Obviously  we  now  have  all  the  factors,  though  our  trial 
division  did  not  proceed  beyond  3. 

"  A  neater  way  of  writing  down  the  prime  factors  of  350  is 

28  x  3a  x  5. 

The  little  3  at  the  top  right-hand  corner  of  the  2  shows 
the  number  of  twos  and  is  called  an  index. 

"  Express  18900  as  the  product  of  factors  which  are  all 
prime. 

18900  =  2x2x3x3x3x5x5x7 
=  22  x  33  X  52  x  7. 

We  read,  2  squared  into  3  cubed  into  5  squared  into  7. 
— An  index  serves  as  a  useful  means  of  shortening  our  written 
work." 

At  this  stage  two  or  three  minutes'  brisk  mental  work 
occasionally  will  help  to  impress  upon  the  pupils'  minds  the 


FACTORS  AND   MULTIPLES  59 

values  of  the  lower  powers  of  the  smaller  numbers:    23,  24, 
25,  32,  33,  34,  42,  43,  44,  52,  53,  54,  &c. 


Common  Factors 

"  Give  me  a  common  factor  of  36  and  48:  2;  another? 
3;  another?  4;  another?  6;  another?  12.  Which  is  the 
greatest  of  these  common  factors?  12.  We  call  12  the  Greatest 
Common  Factor  of  36  and  48.  If  we  write  down  the  prime 
factors  of  the  different  numbers,  we  can  almost  see  the  G.C.F. 
at  once. 

36  =  2x2x3x3. 

48  =  2x2x2x2x3. 

Evidently  2  is  a  common  factor  of  both  numbers,  and 
another  2,  and  a  3.  Hence,  the  G.C.F.  ==  2  x  2  x  3  ==  12, 
i.e.  12  is  the  greatest  number  that  will  divide  exactly  into  36 
and  48." 

"  It  is  neater  to  write  down  the  factors  in  the  index  form. 
—What  is  the  G.C.F.  of  540,  1350,  2520? 

540  =  22  x  33  x  51. 
1350  =  21  X  33  X  52. 
2520  =  23  X  3a  X  51  X  7 

We  see  that  2,  3,  and  5  are  factors  common  to  all  three 
numbers;  from  the  indices  we  see  that  one  2,  two  3's,  and 
one  5  are  common.  Hence  the  G.C.F.  is  21  X  32  X  51  =  90. 
Note  that  we  write  down  each  prime  factor  that  is  common 
and  attach  to  it  the  smallest  index  from  its  own  group. " 

However  clear  the  teaching,  I  find  that  there  is  usually 
a  small  number  of  slow  boys  who  are  puzzled  by  the  index 
grouping.  Hence  in  lower  Sets  the  extended  non-indexed 
groups  of  factors  are  preferable.  Always  sacrifice  a  neat 
method  if  it  leads  to  puzzlement  and  inaccuracy. 


60  CRAFTSMANSHIP  IN  MATHEMATICS 


Common  Multiples 

'  Give  me  a  multiple  of  5:  25;  another?  35;  another? 
55.  Give  me  a  multiple  of  3:  21;  another?  15;  another? 
60.  Give  me  a  common  multiple  of  3  and  5:  60;  another? 
15;  another?  30.  Which  is  the  least  of  all  the  common 
multiples  of  3  and  5?  15;  i.e.  15  is  the  smallest  number  into 
which  3  and  5  will  divide  exactly.  We  call  it  the  Least 
Common  Multiple^ 

"  Find  the  L.C.M.  of  18,  48,  and  60." 

Write  down  the  numbers  as  products  of  their  factors, 
expressed  in  primes. 

18  =  2  X  3  X  3. 

48  —  2x2x2x2x3. 
60  =2x2x3x5. 

The  L.C.M. ,  being  a  multiple  of  the  three  given  numbers, 
must  contain  all  the  factors  of  the  numbers,  but  it  must 
not  contain  more,  or  it  will  not  be  the  least  common  multiple. 
"  1.  The  L.C.M.  has  to  contain  all  the  factors  of  18; 
write  them  down  as  part  of  the  answer: 

L.C.M.  =  2  x  3  x  3  x 

"  2.  In  order  that  the  L.C.M.  may  contain  all  the  factors 
of  48,  it  must  include  four  2's  and  one  3.  We  have  already 
written  down  one  2;  hence  we  must  write  down  three  more. 
As  we  have  already  written  down  two  3's,  another  is  not 
necessary.  Hence, 

L.C.M.  =2x2x2x2x3x3x 

"  3.  In  order  that  the  L.C.M.  may  contain  all  the  factors 
of  60,  it  must  include  two  2's,  one  3,  and  one  5.  We  already 
have  two  2's,  and  a  3,  but  no  5.  Hence  we  must  include 
a  5.  ^ <  ^  ^  ^  ^  ^/ 

L.C.M.  =  2x2x2x2x3x3x5 
=  720. 


FACTORS   AND   MULTIPLES  61 

"  It  is  neater  to  write  down  the  factors  in  the  index  form. 
—What  is  the  L.C.M.  of  54,  72,  240?" 

64  =  2  X  33. 
72  =  23  X  32. 
240  =  24  X  31  X  6l. 

We  mayjvritc  down  the  L.C.M.  at  once^,  by  writing  downjwerx 
one  of  the  prime^  factors  and  attacEmg  to_gach  the  greatest 
index  ot  its  group: 

L.C.M.  =  24  x  3s  x  5 
=  2160 

It  is  well  to  provide  pupils  with  some  little  mnemonic, 
to  enable  them  to  keep  in  mind  that: 

the  smallest  index  concerns  th^ greatesJ^C^. 
and  the  greatest  index  concerns  the  fegg^CJVI^ 

There  is  much  to  be  said  for  using  the  terms  greatest 
and  smallest  (or  least)  in  arithmetic,  and  the  terms  highest 
and  lowest  in  algebra.  The  former  terms  are  obviously 
correctly  applicable  to  magnitudes.  Beginners  do  not 
find  it  easy  to  appreciate  the  exact  significance  of  highest 
and  lowest.  The  term  measure  is  best  avoided.  The  dis- 
tinction between  it  and  factor  is  a  little  subtle  for  boys. 

The  old-fashioned  division  methods  of  G.C.F.  and  L.C.M. 
are  cumbrous  and  unnecessary,  and  slower  boys  never  un- 
derstand the  processes,  the  formal  "  proofs  "  of  which  are 
quite  difficult  enough  for  Fifth  Forms.  Numerical  illustrations 
of  the  principle  that  a  common  factor  of  two  numbers  is  a 
factor  of  their  difference  may  be  utilized  to  justify  the  ordinary 
G.C.F.  procedure,  if  the  procedure  itself  is  considered 
necessary. 


62  CRAFTSMANSHIP  .  IN ,  MATHEMATICS 

CHAPTER  VIII 

Signs,  Symbols,   Brackets.      First 
Notions   of  Equations 

Terminology  and  Symbolism 

Mathematical  terms  should  always  be  used  with  precision; 
then  formal  definitions  in  all  early  work  will  be  unnecessary. 
Sum,  difference,  product,  quotient  are  terms  which  should  be 
quite  familiar  even  to  Juniors;  they  are  the  A,  B,  C  of  the 
whole  subject.  So  should  the  signs  +  ,  — ,  X,  and  — -. 
Multiplicand  and  multiplier,  dividend  and  divisor  should  also 
become  current  coin  at  an  early  stage,  though  there  is  difference 
of  opinion  about  the  first  term  in  this  group.  I  am  not  quite 
sure  about  subtrahend  and  minuend,  even  in  the  senior  school; 
they  are  commonly  confused.  If  we  bear  in  mind  the  English 
significance  of  the  Latin  -nd-  (gerundive),  the  -nd  terms 
can  be  explained  in  a  group. 

Multiplicand,  is  a  number  that  has  to  be  multiplied. 
Dividend  is  a  number  that  has  to  be  divided. 
Subtrahend  is  a  number  that  has  to  be  subtracted. 

And  of  course  minuend  is  a  number  that  has  to  be  minus-ed 
or  reduced,  but  boys  will  confuse  minuend  and  subtrahend. 
If  the  terms  are  used  let  subtrahend  come  first,  and  minuend 
a  good  deal  later. 

As  for  the  division  sign  -f-,  hammer  in  the  fact  that  the 
dots  stand  for  numbers,  that  when  we  write,  e.g.,  |-  we  mean 
4  divided  by  5,  and  that  we  might  write  out  our  division  table, 


either 


5-1  =  5 
10  -f-  2  =  5 
15  ~  3  =  5 


Y  = 


since  both  mean  exactly  the  same  thing. 

Algebraic  letter  symbols  may  be  introduced  at  a  very 


SYMBOLS,  BRACKETS,  EQUATIONS  63 

early  stage.  (Do  not  look  upon  arithmgtjc^giijd  algebra^as 
cUstantjcou§in&,  but  as  twin  brothers,  children  to  be  brought 
up  together.}  Begin  with  the  simple  consideration  of  lengths 
and  areas.  Establish  by  a  few  numerical  examples  that  the 
area  of  a  rectangle  may  be  determined  by  multiplying  length 
by  breadth.  Select  rectangles  whose  sides  are  exact  inch- 
multiples,  ignoring  all  fractions  until  later.  Then  introduce 
the  notion  of  a  "  formula  " — a  convenient  shorthand  means 
of  keeping  an  important  general  arithmetical  result  in  our 
mind.  "  We  have  found  that  however  many  inches  long, 
and  however  many  inches  broad,  a  rectangle  is,  the  area  in 
square  inches  is  equal  to  the  product  of  the  inches  length 
and  the  inches  breadth.  It  is  easy  to  remember  this  by  taking 
the  first  letter  of  the  word  length  (/),  of  the  word  breadth 
(6),  and  of  area  (A),  and  writing  the  result  so: 

/  x  b  =  A. 

But  we  generally  save  time  by  writing  Ib  =  A,  omitting  the 
multiplication  sign.  Always  remember  that  when  in  algebra 
two  letters  are  written  side  by  side,  a  multiplication  sign 
is  supposed  to  be  between  them.  Instead  of  the  letters  /, 
by  and  A,  any  other  letters  might  be  used." 

Rub  in  well  the  principle  taught,  giving  a  few  simple 
evaluations. 

Now  consider  a  square  area,  /  X  /,  or  m  X  m\  II  or  mm. 
(Distinguish  between  inches  square  and  square  inches.) 

Follow  this  up  with  cases  of  rectangular  solids,  and 
establish  such  formulae  as  V  =  I  X  b  X  h  =  lbh\  then  the 
cube,  V  =  aaa. 

"  When  we  were  working  factors,  we  adopted  a  plan  for 
shortening  our  work.  Instead  of  writing  4  X  4  X  4,  we  wrote 
43,  the  little  3  at  the  top  right-hand  corner  (which  we  called 
an  index)  showing  the  number  of  4's  to  be  multiplied  together. 
So  in  algebra. 

aa  may  be  written  a2, 
aaa  may  be  written  03. 

Then  what  does  a5  mean?   a3i2?"   And  so  on. 


64  CRAFTSMANSHIP  IN  MATHEMATICS 

Avoid  all  difficult  examples  at  this  stage.  The  main  thing 
is  to  teach  the  new  principle.  Keep  the  main  issue  clear. 
Let  hard  examples  wait. 

Brackets 

"  A  pair  of  brackets  is  a  sort  of  little  box  containing 
something  so  important  that  it  has  to  receive  special  attention. 
The  brackets  generally  contain  a  little  sum  all  by  itself.  If 
I  write 

9  +  (7  +  3) 
or  9  -f  (7  -  3), 

I  mean  that  the  answer  to  the  little  sum  inside  the  brackets 
has  to  be  added  to  the  9.  If  I  write 

0  -  (6  +  2) 

or  9  -  (5  -  2), 

I  mean  that  the  answer  to  the  little  sum  inside  the  brackets 
has  to  be  subtracted  from  the  9. 

"  Now  I  will  work  out  the  four  sums: 

9  +  (7  +  3)  =  9  +  10  =  19. 

9  +  (7  -  3)  =  9  -f  4  =  13. 

9  -  (5  +  2)  =  9  -  7  =    2. 

9  _  (5  _  2)  =  9  -  3  -    6. 

Are  the  brackets  really  of  any  use?  Let  us  write  the  same 
sums  down  again,  leaving  the  brackets  out,  and  see  if  we 
get  the  same  answer: 

9  4-  7  +  3  =  19 
9  +  7  -  3  =  13 
9  -  5  -f  2  =  6 
9  -  5  -  2  =  2. 

The  first  two  answers  are  the  same,  the  last  two  are  not.  But 
look  at  the  last  two  again.  It  looks  as  if  they  had  been  changed 
over.  Thus 

9  —  (5  +  2)  is  the  same  as  9  —  5  —  2, 
and   9  —  (5  -  2)  is  the  same  as  9  -  5  +  2." 


SYMBOLS,  BRACKETS,  EQUATIONS  65 

With  a  few  easy  examples  like  this,  we  are  in  a  position  to 
justify  the  rule  that  a  -f  sign  before  a  bracket  does  not  affect 
the  +  and  —  signs  within,  but  that  a  —  sign  before  a  bracket 
has  the  effect  or  converting  +  and  —  signs  within  to  —  and 
f~~ respectively^ Thus  you  are  able  to  give  the  rule  and  to 
justifyit!  That  is  enough  at  present.  Give  enough  easy 
examples  to  ensure  that  the  rule  is  known  and  can  be  applied 
with  certainty.  "  Proof  "  should  play  no  part  at  this  early 
stage.  The  algebraic  minus  sign  comes  later. 

Now  show  the  effect  of  a  multiplier. 

4(6  -f-  3)  =  4  X  9  ==  36. 

"  We   might  have   multiplied   the  two   numbers   separately 
in  this  way, 

4(6  +  3)  -  24  +  12  =  36,* 

and  when  the  brackets  contain  both  letters  and  numbers  we 
must  do  it  in  that  way: 

5(N  -f  3)  =  5N  +  16 
for  we  cannot  add  N  to  3." 


First  Notions  of  Equations 

Simple  equated  quantities. — For  convenience  at  this  early 
stage  we  may  call  the  following  an  equation: 

7  +  5  =  21  -  9. 

Establish  the  fundamental  fact  about  an  equation  that  we 
may  add  to,  subtract  from,  multiply,  or  divide  each  side  of 
an  equation  by  any  number  we  like,  provided  that  we  use 
the  same  number  for  both  sides.  Give  several  examples, 
to  illustrate  each  of  the  four  operations.  To  enable  the  class  to 


*  A  repetition  of  the  sign  of  equality  in  the  same  line  should  never  be  allowed 
in  school  practice;  it  is  almost  always  ambiguous.  We  do  it  sometimes  in  this  book 
merely  to  save  space. 

( E  291 )  6 


66  CRAFTSMANSHIP  IN  MATHEMATICS 

see  the  operations  more  clearly,  put  the  original  quantities 
in  brackets.   Thus: 

(7  +  5)  +  4  =    (21  -  9)  +  4 
(7  +  5)  -  4  =    (21  -  9)  -  4 
4(7  +  5)          =  4(21  -  9) 
7  +  5  21-9 


Do  not  talk  about  "  proofs  ";  you  are  merely  verifying 
particular  instances,  to  enable  the  boys  to  see  that  your  rules 
are  not  arbitrary  but  are  based  on  reason.  A  little  practice 
in  such  easy  examples  as  the  following  may  usefully  follow. 

5  times  a  certain  number  is  65.     What  is  the  number? 

"  We  have  to  find  a  certain  unknown  number.  Let  us  call 
it  N.  The  sum  tells  us  that 

5N  ==  65. 

Divide  each  side  of  the  equation  by  5;  then,  N  =  13,  the 
number  we  require." 

The  class  is  not  quite  ready  for  such  an  example  as  the 
following,  but  they  can  follow  out  their  teacher's  reasoning, 
and  their  appetite  is  whetted. 

Divide  32  into  two  parts,  so  that  5  times  the  smaller  is 
3  times  the  greater. 

"  The  two  parts  added  together  must  make  32,  so  that 
one  part  taken  from  the  32  must  give  the  other. 

Let  S  stand  for  the  smaller  number. 
Then  32  —  S  must  represent  the  greater. 
The  sum  tells  us  that 

6  times  the  smaller  ==  3  times  the  greater. 

So  we  may  write 

6S  =  3(32  -  S). 


SYMBOLS,  BRACKETS,  EQUATIONS  67 

Removing  the  brackets,  by  multiplying  by  3, 
5S  =  96  -  3S. 

We  cannot  see  the  value  of  S  from  this,  because  we  have  S's 
on  both  sides  of  the  equation.  But,  adding  3S  to  each  side, 
we  have: 

6S  +  3S  ==  96  -  3S  +  3S 

8S  -  96 

S  =  12,  the  smaller  number 
and    32  —  S  =  20,  the  greater  number. 

Now  let  us  verify  the  results;"   &c. 


CHAPTER   IX 

Vulgar   Fractions 

First  Notions  of  Fractions 

Vulgar  or  Decimal  Fractions  first?  The  first  notions  of 
vulgar  fractions  will  be  given  in  the  preparatory  Forms, 
where  the  significance  of  at  least  halves  and  quarters  will  be 
understood  and  the  manner  of  writing  them  down  known. 
In-  the  lower  Forms  of  the  senior  school,  it  is  probably  wise 
first  to  give  a  few  lessons  on  the  nature  and  manipulation  of 
vulgar  fractions,  then  to  proceed  with  decimals,  and  to 
return  to  the  more  difficult  considerations  of  vulgar  fractions 
later. 

The  first  thing  is  to  get  clearly  into  the  child's  mind 
that  a  fraction  is  a  piece  of  a  thing,  a  piece  "  broken  off  "  a 
thing.  Take  one  of  several  similar  things  (sticks,  apples), 
and  break  off  or  cut  off  a  "  fraction  "  of  it.  Cut  one  of  the 
things  into  2  equal  parts,  and  introduce  the  term  halves; 
into  3,  and  introduce  the  term  thirds;  into  4,  and  the  term 


68  CRAFTSMANSHIP  IN  MATHEMATICS 

quarters;  and  see  that  the  terms  halves,  thirds,  quarters, 
fifths,  &c.,  are  made  thoroughly  familiar.  "  I  have  cut  this 
apple  into  8  parts:  give  me  1  eighth;  give  me  5  eighths.  I 
add  2  of  the  eighths  and  5  of  the  eighths  together:  how  many 
eighths  have  I?" 

We  have  a  special  way  of  writing  down  fractions.  We 
draw  a  line;  under  it  we  write  the  name  of  the  parts  we  cut 
the  apple  into,  over  it  we  write  a  figure  to  show  the  number 
of  the  parts  we  take:  thus 

3  4 


fifths '     sevenths 

Parts  of  the  same  name  may  be  added  together.    Just  as  we 
say 

2  apples  +  3  apples  =  5  apples, 

so  we  may  say, 

2  sevenths  -}-  3  sevenths  =  5  sevenths, 

and  we  write, 

23  5 


sevenths       sevenths       sevenths 

Let  the  child  see  clearly  that  the  fraction  shows 

number  of  parts 
name  of  parts  ' 

and,  a  little  later  on,  introduce  the  terms  numerator  and 
denominator: 

numerator 
denominator' 

where  num  =  number  and   nom  =  name.      If  the   children 
learn  Latin,  give  the  Latin  words. 

Then  come  to  fractions  of  collections  of  things:  |  of  the 
class  of  children,  J  a  basket  of  apples,  -^  of  a  Ib.  of  cherries. 
The  way  is  now  paved  to  fractions  of  mere  numbers:  £  of  32; 
^  of  27;  and  so  on.  But  at  this  stage  avoid  the  terms  abstract 
and  concrete. 


VULGAR  FRACTIONS  69 

For  illustrating  fractional  processes,  every  teacher  will 
utilize  concrete  examples  of  some  kind  drawn  from  everyday 
lifeT  "As  the  number  60  contains  numerous^  easy  factors, 
fractions  of  a  crown  (60rf.)  and  of  an  hour  (60  minutes) 
make  good  examples  for  mental  work. 

Mental  work  may  profitably  be  undertaken  as  soon  as 
the  nature  of  a  fraction  is  fully  grasped. 

"  Number  of  pence  in  |  of  I/-?  in  §?  in  |?  in  |?  in 
yr2-?  in  ^  ?"  Let  the  children  thus  discover  that  f  =  ^  -  =  -j^-, 
that  different  fractions  may  therefore  have  the  same  value. 
Thus  we  come  to  the  notion  of  "  cancelling  "  and  its  con- 
verse. 

Again: 

"  How  many  minutes  in  -3-  of  an  hour?     20. 

i  ?     1  r) 

»  »  »  4  »  »     •          •*•'•'• 

1  ?      10 

»  »  »  5  »  »     •  • 

Thus  (5  +  J  +  i)  ot  an  hour  —  47  minutes. 
How   may   we   express   47    minutes   as   the   fraction   of  an 

.'.  *  +  i  +  2  =  U- 


hour? 


Can  you  see  that  this  is  true?    No. 

"  Well,  we  have  seen  that  }  of  an  hour  =  20  minutes, 
and  since  1  minute  ---=  (.](j  of  an  hour,  20  minutes  =  ;?§  of 
an  hour;  &c.  Hence  we  may  write  ^  +  \  +  -^  in  this  way: 

•eS  +  IS"  +  ^ 

and  wow  it  is  easy  to  see  why  the  answer  is  ^-. 

"  Thus  if  we  want  to  add  fractions  together,  w^jaust 
firstsee  that  they  are  fractions^ol  the  same  name,  i.e.  that 
they  have  the  same  denominator. 

"  But  how  are  we  to  change  fractions  of  different  denom- 
inators to  fractions  of  the  same  denominator  ?"  —  And  sojvve 
cometo  L.CJVI.s,  &c. 

On"  the"  whole,  however,  I  prefer  to  illustrate  fractional 


70 


CRAFTSMANSHIP  IN  MATHEMATICS 


processes  by  means  of  diagrams,  rectangles  rather  than  lines. 
A  rectangle  is  conveniently  divided  up  into  smaller  rectangles 
by  lines  drawn  in  two  directions,  and  thus  the  fraction  of 
a  fraction  is  easily  exhibited.  We  append  a  few  diagrams. 
(A  squared  blackboard  or  squared  paper  is  always  advisable.) 

DIAGRAMMATIC  ILLUSTRATIONS 
Cancelling. —  -^  =  f  =  f . 


X* 


Addition.-  J  +  i  +  |  =  A  +  A  +  A  = 


M 


* 


* 


Fig.  2 

Subtraction. —  f  —  -/2-  =  -f$. 

A E 


Fig.i 

These  figures  illustrating  subtraction  will  puzzle  the  slower 
children,  but  squared  paper  and  scissors  will  soon  help  to 
make  things  clear. 

Multiplication 

Multiplication  by  a  fraction  is  always  a  little  puzzling 
at  first.    A  child  naturally  expects  a  multiplication  sum  to 


VULGAR  FRACTIONS 


71 


produce  an  answer  bigger  than  the  multiplicand.    It  is  best 
to  begin  with  mixed  numbers. 

The  child  knows  that  2/-  X  3  is  6/-;  by  4,  is  8/-;  and  by 
3£,  is  7/-.  Hence  multiplying  by  the  £  seems  to  him  some- 
how to  have  been  a  real  multiplication,  inasmuch  as  the  6/- 
has  been  increased  to  7/-.  The  multiplication  may  be  con- 
sidered, as  usual,  as  an  addition,  viz.  of  3  florins  and  a  half- 
florin.  Multiplying  a  florin  by  |  is  to  take  the  half  "  of  "  a 
florin.  Give  other  examples  to  show  clearly  the  meaning  of 
the  word  "  of  "  when  we  speak  of  multiplying  by  a  fraction. 

Example:   Multiply  2  sq.  in.  by  3|. 


Fig.  4 

The  figure  shows  2  sq.  in.  one  above  the  other,  then 
another  2  sq.  in.,  then  another,  then  \  part  of  2  sq.  in.  The 
last  piece  shows  multiplication  by  a  real  fraction,  viz.  2  in.X^, 
i.e.  the  strip  is  \  of  2  sq.  in.  Thus  the  whole  figure  is  6§ 
sq.  in.  Hence  2  x  3J  =  6|. 

Another  example:  Draw  a  figure  to  show  3J  sq.  in.\  then 
show  this  multiplied  by  2J. 


S'4  5?  In 

9. 

, 

....« 

SyJe*.'. 

.?*-.-.- 

• 

r          •' 

Fig.  5 


72  CRAFTSMANSHIP  IN  MATHEMATICS 

"  How  from  these  figures  may  we  obtain  an  answer  to  the 
sum  3J  x  2|?  Let  us  first  think  of  money,  say  shillings, 
instead  of  inches. 

3Js.  =  3s.  3d.  =  39rf. 

Twice  39<*.  =  78^.;  one-third  of  39J.  =  I3d. 
/.     2J  times  39J.  =  (78  +  13) J.  =  9 Id.  =  7*.  7d.  =  7^5. 

Apparently,  then,  3^  X  2|  =  7-112-.  Thus,  if  we  divide  the 
last  figure  up  into  twelfths,  we  ought  to  see  whether  this 
result  is  true.  Fig.  6  shows  that  it  is  true;  the  squares  need 
not  be  true  squares;  oblongs  will  show  the  fractions  just  as 
well. 


Fig.  6 

Number  of  twelfths  in  the  figure: 

In  the  6  big  squares  or  oblongs,  12  each  =  72 

In  the  2  strips  at  the  sides,  3  each  =     6 

In  the  3  strips  at  the  bottom,  4  each  =  12 

In  the  small  strip  at  the  corner  =     1 

Total  91 

Thus  in  the  figure  we  have  91  twelfths  =  f|  =  7£,  as 
expected." 

Such  a  figure  is  satisfactory  for  illustrating  the  multiplica- 
tion of  mixed  numbers,  but  for  multiplication  purposes 
mixed  numbers  should  rarely  be  turned  into  improper 
fractions.  A  different  example  is  therefore  advisable. 

Draw  a  figure  to  show  f  of  |.  Lead  up  to  the  necessary 
figure  by  showing,  first,  f;  then,  by  dividing  the  thirds  into 


VULGAR  FRACTIONS 


73 


Fig.  7 

7  parts,  show  y  of  \\  lastly,  show  |  of  f.     Obviously,  now, 

4    v    2    __  _8 

7   *   :i  "~  21"' 

"  We  have  found  out  that: 

(i),  2  x  3J,  or  {  X  7,  -  2^°; 
and  (ii),  3]  x  2£,  or  -\;J-  X  J,  =  *?-; 
and  (iii),  i  X  }=  /t. 

Now  look:  in  every  case  the  numerators  multiplied  together 
give  the  numerator  in  the  answer,  and  the  denominators 
multiplied  together  give  the  denominator  in  the  answer." 

Now  the  teacher  is  in  a  position  to  enunciate  the  rule. 
He  has  done  nothing  to  prove  the  rule,  but  he  has  justified 
it,  so  far  as  it  can  be  justified  with  beginners. 

Division 

Just  as  a  child  naturally  expects  a  multiplication  always 
to  produce  an  increase,  so  he  expects  a  division  always  to 
produce  a  decrease. 

"  When  you  have  divided  a  number  by  another,  the 
dividend  is  always  made  smaller.  Do  you  agree?" — Yes. 
"  Always  smaller?"— Yes.  "  Quite  certain?"— Yes. 

"  Let  us  divide  36  pence  amongst  some  boys. 

I  give  them  12 d.  each:  how  many  boys?      3. 


od. 
2d. 
Id. 


12. 

18. 
ob. 


Do  you   mean  to  say  I   cannot  divide  the   36  pence 


74 


CRAFTSMANSHIP  IN  MATHEMATICS 


amongst  more  than  36  boys?" — Yes,  if  you  give  them  less 
than  Id. 

"  Then  give  them  %d.  each.    How  many?"   72. 

"  Then  36  divided  by  J  is  72.  Thus,  although  I  have 
divided  36  I  have  a  quotient  bigger  than  36.  So  you  were 
wrong!"  And  so  on. 

A  suitable  scheme  of  diagrammatic  division  is  easily 
devised,  but  it  is  best  approached  by  the  division  of  whole 
numbers. 

When  we  divide  24  by  4,  the  quotient  is  6,  and  the 
4  sixes  may  be  arranged  in  4  lines  thus: 


Fig.  8 


The  shaded  section  denotes  the  quotient  (6  units);  it  is  a 
row  of  units  in  line  with  1  of  the  units  of  the  divisor,  4.  Any 
other  row  would  have  done  equally  well,  for  any  other  row 
would  have  been  in  line  with  1  of  the  units  of  the  divisor,  4. 

So  generally;  a  rectangle  representing  the  dividend  may 
always  be  divided  up  in  such  a  way  that  each  horizontal  row 
of  units  represents  the  quotient;  there  are  as  many  horizontal 
rows  as  there  are  units  in  the  divisor.  Opposite  any  unit 
of  the  divisor  (we  select  the  first)  is  a  horizontal  row  of  units 
representing  the  quotient. 

We  show  12  divided  by  2,  by  3,  and  by  4: 


Fig.  9 


VULGAR  FRACTIONS 


75 


Let  the  first  fractional  problem  be  to  divide  4£  by  1£. 
We  may  ask  how  many  times  I^s.  is  contained  in  4£s.,  i.e. 
how  many  times  16d.  is  contained  in  54d.  We  may  show 
this  division  in  the  ordinary  way,  |-f  or  -8^,  which  is  equal 
to  3f .  Thus  the  answer  to  the  sum  is  3$.  How  are  we  to  show 
this  in  a  diagram? 

We  will  first  divide  3,  represented  by  a  rectangle  of  3  sq.  in., 
by  4,  by  3,  by  2£,  by  2,  by  1J,  by  1,  by  f. 


3-r  4=  } 


3  -  2J  =  | 

(Divide  AB  into  5,  and  take 

2  parts  for  the  unit.) 


3  -7-  1J  =  -J  3^1  =  3 

(Divide  AB  into  3,  and  take 
2  parts  for  the  unit.) 

Fig.  10 

Examine  the  six  diagrams,  and  note  how  the  quotient  (the 
shaded  part)  increases  as  the  divisor  diminishes.  If  then 
we  diminish  the  divisor  further,  the  quotient  (the  shaded 
part)  must  be  still  bigger.  As  before,  the  shaded  quotient 
must  occupy  a  space  opposite  one  complete  unit  of  the 
divisor.  But  in  this  case  AB  is  not  long  enough  to  show  a 


76  CRAFTSMANSHIP  IN   MATHEMATICS 

complete  unit,  only  |-  of  one.   So  we  must  extend  it  to  make 


3.      4    16 
—  7f  —  -4- 


Fig.  1 1 


We  may  now  return  to  our  original   example:     Divide 
by  1J. 


Fig.  12 

One  more  example:    Divide  f  fry  f . 

A 


Fig.  13 

The  first  figure  shows  1  sq.  in. 

The  second  figure  shows  f  sq.  in.,  the  part  to  be 

divided  by  f . 
The  third  figure  shows  AB  extended  to  one  complete 

unit,  since  AB  itself  represents  only  f  of  one. 
The  last  figure  shows  the  result  of  the  division,  viz. 

§.     3   8 
~   4  —  ^9- 


VULGAR  FRACTIONS 


77 


We  may  now  collect  up  our  results: 


5  fl 

•2    =    5- 


4  __  1  5_ 

4  ii  7 

;3  ~~  "s" 

3  _  « 

4  1» 


An  average  class  will  soon  discover  that  by  inverting  the 
divisor  the  quotient  is  then  obtained  by  treating  the  sum 
as  a  multiplication  sum,  e.g. 


But  let  the  teacher  be  under  no  delusion.  Only  a  very  small 
minority  will,  at  the  time,  appreciate  the  purpose  of  the 
diagrams.  Over  and  over  again  I  have  seen  a  majority  com- 
pletely baffled,  even  with  very  skilful  teaching.  No  matter. 
Come  back  to  the  demonstration  again,  a  year  or  two  later. 
You  have  justified  your  rule  as  far  as  you  can.  Now  state 
it  in  clear  terms  and  —  for  the  present  —  be  satisfied  that  the 
boys  are  able  to  get  their  sums  right.  There  is  probably 
nothing  more  difficult  in  the  whole  range  of  arithmetic 
than  the  division  of  fractions,  i.e.  for  boys  to  understand 
the  process  when  it  is  first  taught. 

The  following  kind  of  argument  is  sometimes  useful: 
To  divide  a  fraction,  say  4|,  by  5  is  the  same  thing  as 
taking  £  of  4|.    But  to  take  I  "  of  "  4£  is  the  same  thing 
as  multiplying  4|  by  -*-;   i.e. 

4J  +  5  =  4J  x  }. 

Now,  if  we  divide  4|  by  ^,  we  divide  it  by  a  number  7  times 
as  small  as  when  we  divided  by  5;  therefore  our  answer 
must  be  7  times  as  large  as  before;  i.e. 

Since  4J  ~-  5  =  4£  x  -J- 


i.e   to  divide  by  |-  is  the  same  thing  as  multiplying  by  -J. 
Hence,  once  more,  the  rule  of  inverting  the  divisor. 

But  the  argument  is  quite  beyond  the  average  beginner, 
as  every  experienced  teacher  knows. 


78  CRAFTSMANSHIP  IN  MATHEMATICS 

Cancelling.  —  When  a  boy  is  told  he  may  "  cancel 
thus: 

2       5 


he  is  likely  to  ask,  why?    He  will  already  have  learnt: 

(1)  That,  e.g.,  7  X  9  -  9  X  7; 

(2)  Reduction  of  fractions  to  their  lowest  terms; 

(3)  Multiplication  of  fractions. 

Thus  he  will  understand  that 

If  x  —  =  44  x  25 

46        22  ~~  45  X  22  ' 

and  that  this  may  be  written: 

44  X  25  44        25 

_  _  __  QJi  _          ^        _ 

22  x  45  22        45' 

He  now  readily  sees  that  he  is  justified  in  reducing  each 
of  these  to  its  lowest  terms,  and  that  the  final  result  is  the 
same  as  when  he  cancelled  terms  in  different  fractions. 


CHAPTER  X 

*  t 

Decimal   Fractions 

A  Natural  Extension  of  Ordinary  Notation 

If  care  is  taken  to  teach  the  inner  nature  of  decimal 
notation  thoroughly,  decimals  need  present  little  difficulty. 

"  A  decimal  fraction  is  merely  a  particutyr  kind  of  vulgar 
fraction,  viz.  one  with  a  denominator  10  or  power  of  10, 


DECIMAL  FRACTIONS  79 

e-g-  Tb-»  T<io>  Hoo>  Toto  o-  But  we  do  not  generally  write 
them  this  way;  we  write  them  as  follows: 

•7,  -73,  8-192,  -0003 

"  Let  us  consider  a  number  consisting  entirely  of  ones 
(any  other  figure  would  do  equally  well). 

The  number        11  =  10  +  1 

The  number      111  =  100  +  10  +  1 

The  number    1111  -  1000  +  100  +10  +  1. 

The  1  of  least  importance  in  each  number  is  the  1  on  the 
extreme  right;  each  1  to  the  left  is  10  times  as  important 
as,  and  is  10  times  the  value  of,  its  right-hand  neighbour. 
The  1  on  the  extreme  right  represents  just  one  unit. 

"  But  we  often  break  up  a  unit  into  parts,  e.g.  a  sovereign, 
or  a  bag  of  nuts.  These  parts  are  fractions,  and  we  might 
carry  our  ones  to  the  right,  to  represent  these  fractions, 
devising  some  means  of  separating  the  fractions  from  the 
whole  numbers:  a  straight  line  would  do. 

Thus,     1111|1  may  stand  for  1000  +  100  +  10  +  1|  +  ^ 
and,     11|111  may  stand  for  10  +  1|+  ^  +  t  ^  + 


Any  sort  of  separating  mark  will  do.  Generally  we  use  the 
smallest  possible  mark,  a  dot,  written  half-way  up  the  height 
of  the  figures. 

"  Thus  276-347  -  200  +  70  +  6  +  -^  +  jfo  +  T^. 
If  we  add  the  3  fractions  together,  we  get  -nrcnr     Thus 


276-347  ~  276  v 

"  Note  the  two  ways  of  writing  down  the  same  thing: 

~ 


_      4      _L_       5          i  0  i  1  _    4501 

—  i  o  ~r  TOCT  ~r  Twer  ~r  ~ 


__  _ 

i  o  ~r  TOCT  ~r  Twer  ~r  unnnr  ~  ioooo 

"  Perhaps    there    are    no    tenths,    and    we    begin    with 
hundredths: 

•042  = 


We  could  not  write  this  -42  because  -42  —  Y 


80 


CRAFTSMANSHIP  IN  MATHEMATICS 


Thus  we  may  have  one  or  more  noughts  between  the  decimal 
point  and  the  "  significant  "  figures. 

"  But  noughts  on  the  extreme  right  hand  of  a  decimal 
have  no  meaning: 

•034  =  A  +  T? 

1  0     '     1 i 

as  before/' 

Give  ample  practice  in  conversion  and  reconversion,  until 
the  change  can  be  written  down  mechanically: 

37-063  ==  37^^;     ^g^  r-  -0421;     &c.,  &c. 
'conversion  of  either  decimal  form  into  the  other  is  essential. 


4         JJ4 

iodo        1000 

T(JCT(J  + 


o        _   34 

1(7(50(1(1  ~  fooo 


This  is  tfie  key  to  a 


Let  the  rules  for  conversion  be  stated   in  the  simplest 

possible  words. 

A  few  exercises  of  the  following  nature  are  useful: 

"  If  x  —  10,  find  the  value  of  these  expressions,  writing 

down  the  answers  in  both  decimal  forms  (do  not  cancel  as 

you  would  vulgar  fractions): 

3  +  -  +  -?2  =  34-  ,40  +  Jff  =  3tVo,  or  3-47. 
x       x2 

6r>  +  4*  +  9  -j-  ~  +  ~  =  6049  1  g-gg0,  or  5049-0306. 


Multiplication  and  Division  by  10,  by  100,   &c. 

"  If  we  multiply  347  by  10  we  obtain  3470,  the  7  becoming 
70,  the  40  becoming  400,  and  the  300,  3000. 


Th. 

H. 

T. 

u. 

3 

4 

7 

3 

4 

7 

0 

Every  figure  is  moved  one  place  to  the  left,  and  its  value  is 
increased  10  times. 


DECIMAL  FRACTIONS 


81 


"  If  we  divide  2180  by  10  we  obtain  218,  the  2000  be- 
coming 200,  the  100  becoming  10,  and  the  80,  8. 


Th. 

H. 

T. 

u. 

2 

1 

8 

0 

2 

1 

8 

Every  figure  is  moved  one  place  to  the  right,  and  its  value 
is  diminished  10  times. 

"  So  it  is  with  decimal  fractions,  or  decimals  as  we  often 
call  them." 


Multiply  3*164  by  10. 

3-164  x  10  =  (3  +  tV  +  T£n  + 
=  30+1-1-  ;'0  +  1( 
=--  31  fVo  -31-64; 


X  10 


i.e.  the  decimal  point  has  been  moved  one  place  to  the  right, 
and  every  figure  occupies  a  place  10  times  as  important  as 
before. 

So  5-623  X  100       =  562-3; 

•005623  x   10000  =  56-23. 


+ 


Divide  3-164  by  10. 

3-164  X£  =  (3+  ,',-,  +  , 
=  >3<>  +  i  w  + 

'M  (14       _ 

—    1  O  0  0  (7  — 

i.e.  the  decimal  point  has  been  moved  one  place  to  the  left, 
and  every  figure  occupies  a  place  reduced  in  importance 
10  times. 

Give  a  number  of  varied  examples  in  both  multiplication 
and  division,  and  help  the  pupils  to  deduce  the  rules. 

Give  plenty  of  mental  work  of  the  following  kind: 

tens  X  tens  =  hundreds 

hundreds  X  tens  =  thousands 

hundreds  X  hundreds  ==  ten  thousands 
tenths  X  tenths  =  hundredths 

hundred ths  X  tenths    =  thousandths. 
(B  291 )  7 


82  CRAFTSMANSHIP  IN  MATHEMATICS 

Continue  this  kind  of  work  until  instant  reponse  is  obtained 
as  to  the  significance  of  moving  the  decimal  point  so  many 
places  to  the  right  or  so  many  to  the  left.  Let  the  notation 
be  mastered;  then  the  rest  will  give  little  trouble. 

Addition  and  Subtraction 

Do  not  forget  the  common  cause  of  inaccuracy,  blanks 
in  the  fractional  columns,  especially  if  the  numbers  are 
arranged  horizontally,  e.g.  7-612  +  3-1  -f  2-0151. 

Multiplication 

72-314  X  -32 

_   79  31  4      vy      32 

~~ 


__    7  'J  3  1  4     v      3  2 
--  TT700      X    lOtf 
_    2  3  1  4  0  t  H 
--  10  (TO  GO 

=  23-14048. 

The  whole  process  resolves  itself  into  (1)  conversion,  (2) 
multiplication  of  whole  numbers,  (3)  reconversion. 

Note  that  "  conversion  "  does  not  mean  conversion  to 
vulgar  fractions,  but  to  the  alternative  form  of  decimal 
fractions,  with  denominators  consisting  of  powers  of  10. 

The  multiplication  of  the  denominator  is  really  nothing 
more  than  the  mere  addition  of  noughts,  and  it  is  obvious 
from  this  multiplication  that  the  number  of  decimal  places 
in  the  product  is  equal  to  the  sum  of  the  numbers  of  decimal 
places  in  the  multiplicand  and  multiplier,  and  that  from  the 
very  nature  of  the  case  this  must  always  be  so. 

Hence  the  simple  rules: 

1.  Ignore  the  decimal  point  and  perform  the  multiplication 
as  if  the  multiplicand  and  multiplier  were  whole  numbers. 

2.  Add  together  the  decimal  place  in  the  multiplicand  and 
multiplier;    this  gives  the  number  in  the  product.     Fix  the 
point  by  counting  back  that  number  of  places  from  the 
right. 


DECIMAL  FRACTIONS  83 

72-314 
•32 

144628 
21 6942 
23-14048"    No.  of  dec.  places  =  (3  -f  2)  =  5 

Is  the  method  intelligent^  It  is  at  least  as  intelligent  as 
any  other  method,  and  it  has  this  advantage — that  the  boy 
works  his  sums  exactly  as  he  works  ordinary  simple  mul- 
tiplication. And  the  procedure  is  easily  and  immediately 
justified,  by  conversion  and  reconversion. 

Give  other  examples,  using  the  same  numbers  but  changing 
the  position  of  the  decimal  points.  The  answers  shall  be 
given  mentally  and  at  once: 

•72314  X  3-2         =  2-314048. 
•0072314  X  32          =  0-2314048. 
72314  X  -00032  =  23-14048. 
723-14  X  320        =  7231-4  X  32 
=  231404-8 

If  preferred,  the  boy  might  set  out  his  working  thus: 
72-314  x  -32  =  ^£4  x  ^ 

2  31  4  0  4  R 

—  ~iTyfrofy<r 
=  23-14048 

and  show  his  actual  multiplication  neatly  on  the  left. 
Possible  objections  to  the  method. 

1.  "  The  most  important  digit  in  the  multiplier  is  not 
used  first." — Granted.     But  this  disadvantage  is  outweighed 
by  the  advantage  of  greater  accuracy. 

2.  "  The  decimal  points  are  not  kept  in  a  vertical  column." 
— This  is  of  no  material  consequence,  though  it  is  quite  easy 
to  teach  the  boy,  if  it  is  thought  worth  while,  to  place  the 
points  in  the  successive  products.    For  instance: 

72-314 

-32 

1-44628 
21-6942 
23-14048 


84  CRAFTSMANSHIP  IN  MATHEMATICS 

The  boy  multiplies  through  by  2  and  then  says,  "  When  I 
multiplied  4  by  2,  I  multiplied  thousandths  by  hundredths, 
and  this  gives  me  hundredths  of  thousandths,  which  occupy 
the  5th  decimal  place;  therefore  the  point  goes  in  front  of 
the  first  4."  He  argues  similarly  when  he  has  multiplied  by 
the  3,  though  he  would  soon  learn  that  the  position  of  the 
point  in  the  first  partial  product  gives  the  key  to  its  position 
in  all  the  other  products.  Thenceforth  he  would  work 
mechanically.  Does  not  the  time  come  when  we  all  work 
mechanically  in  all  types  of  calculation?  does  not  the  rationale 
of  procedure  tend  to  fade  away,  until  something  turns  up 
demanding  revivification? 

Is  there  a  more  intelligent  plan  than  teaching  the  boy  to 
complete  the  actual  multiplying  before  considering  the 
decimal  point  at  all?  I  doubt  it.  And  I  am  quite  sure  that 
no  other  plan  is  productive  of  a  greater  degree  of  accuracy. 
The  boy  has  confidence  in  a  method  so  closely  akin  to  one 
with  which  he  is  already  familiar. 

Standard  Form 

It  has  been  gravely  said  that  "  standard  form  "  was  the 
invention  of  the  devil.  In  reality  it  was  not  quite  so  bad  as 
that.  It  was  invented*  by  an  old  personal  friend  of  my  own, 
the  senior  mathematical  master  of  one  of  our  great  Public 
Schools,  who  decided  that  he  "  must  adopt  some  new  method 
to  prevent  his  boys  from  getting  so  many  sums  right,  in  order 
to  take  the  conceit  out  of  them  ". 

Why  are  the  apologists  of  the  method  always  so  faint- 
hearted? 

*  The   method   was   suggested   by   '*  standard  "   form  in   logarithms,  where  of 
course  it  is  very  useful. 


DECIMAL  FRACTIONS  85 


Division 

As  division  is  the  reverse  process  of  multiplication,  the 
analogous  method  for  fixing  the  decimal  point  may  be  adopted. 

23-14048  -f-  -32 

_    2  :i  1  4  O  4  N 

— 


_    2  .'I  1  4  0  4  H 

----  a  a 


=  72314  x 

__    7  2  :)  1  4 
--  1  006" 

=  72-314. 


The  actual  simple  division  by  32  may  be  neatly  shown  to  the 
left. 

The  simple  rules  are: 

1.  Ignore   the   decimal   points   and   divide   as   in  simple 
division. 

2.  Subtract  the  number  of  decimal  places  in  the  divisor 
from  the  number  in  the  dividend.     This  gives  the  number 
in  the  quotient.    Fix  the  point  by  counting  back  this  number 
from  the  right. 

The  whole  process  resolves  itself  into  (1)  conversion, 
(2)  division  by  whole  numbers,  (3)  reconversion. 

Again:  is  the  method  intelligent?  Again  the  answer  is  that 
it  is  as  least  as  intelligent  as  any  other  method,  and  it  certainly 
makes  for  accuracy.  Here  is  an  example  with  the  working 
as  commonly  shown:  Divide  2-0735  by  8-72. 

_  23 
8-72)2-0735 
1  744 

~3295         Decimal  places  =  4  —  2  =  2. 
2616        Thus  the  quotient  is  "23,  and  a  remainder. 
1379 

The  division  may  now  be  continued  to  any  number  of  places. 

If,   before   dividing,   we   add    O's    to  the   dividend   and 

continue    the    dividing    further,   this    does    not    affect    the 


86  CRAFTSMANSHIP  IN  MATHEMATICS 

decimal    point   in    the    quotient:    e.g.    divide    2-073500    by 
8-72. 

2377 

8-72)2:073500 
1  744 
3295 
2616 


6790          Decimal  places  =6-2  =  4. 

6104          Thus  the  quotient  is  '2377,  and  a  remainder. 

~6860 

6104 

~756 

Hence,  if  a  given  dividend  contains  a  smaller  number  of 
decimal  places  than  the  divisor,  add  O's  to  make  the  number 
equal  (and  more  if  necessary).  Example:  divide  -001  by 

7-0564.    Write:  

7-0564>OOl 

We  cannot  proceed  with  the  division  until  we  add  at  least 
5  more  O's. 

1 
7-0564y00100000 

70564          Decimal  places  =  8-4=4 
29436 

Thus  the  answer  is  '0001  .  .  .  and  a  remainder.  The  quotient 
can  be  carried  to  as  many  places  as  may  be  required. 

The  value  of  the  remainder. — It  is  desirable  to  make 
the  abler  boys  see  the  real  value  of  the  quantities  in  the 
successive  steps  of  the  division.  Example:  divide  '07925 
by  3-7. 

214 

3-7)-07925 
74 
62 

37          Decimal  places  =  5—1  =  4. 
155        Quotient  =  '0214  and  a  remainder. 
148 
7 


DECIMAL  FRACTIONS  87 

What  is  the  value  of  the  "  74  "  in  the  first  step?  It  is  the 
product  of  3-7  and  -02  (as  we  may  see  from  the  quotient), 
and  must  therefore  contain  (1  +  2  or)  3  decimal  places; 
hence  its  value  is  '074.  Similar  arguments  apply  to  each 
step.  Hence,  more  correctly,  the  division  may  be  set  out  in 
this  way: 

•0214 
3-7JM37925 

•074  =  3-7  X  -02 

•0052 

•0037  =  3-7  X  -001 

•00155 

•00148  =  3-7  X  -0004 


•00007 

Thus  the  quotient  (to  4  figures)  is  '0214,  and  the  remainder 
is  *00007.  The  abler  boys  will  soon  learn  to  assign  the 
correct  value  to  the  remainder,  by  merely  glancing  at  the 
dividend  vertically  above  it. 

Verification  should  be  encouraged: 

Dividend  =  (quotient  X  divisor)  +  remainder 
=  (-0214  X  3-7)  +  -00007 
=  -07918  +  -00007 
=  -07925 

Practice  in  manipulation  of  the  following  kind  is  useful: 

3-204  =   3204  =   -03204     =   320-4 
^0701   ~~  ~7(M   ~   -000701   ~     7-01  ' 

The  boy  sees  at  once  that  the  same  quotient  must  result 
from  all  the  division  sums.  The  only  real  defence  for  the 
reduction  of  the  divisor  to  a  form  approximating  a  small 
whole  number  is  that  it  enables  a  boy  to  obtain  a  rough 
answer  by  easy  calculation.  For  instance,  in  the  last  of  the 

r  r  1.  3204          320  ,  .      ,  , 

four  forms  above,  -=-^r-  =  -=~  approximately,  and  thus  the 


88  CRAFTSMANSHIP  IN  MATHEMATICS 

answer  to  this  group  of  division  sums  is  roughly  y  of  320, 
i.e.  a  number  between  40  and  50. 

.    '.         -000983         -0983  -983 

Again:     -^-  =   —     or     _ . 

Thus  the  answer  is  roughly  ~4~7-  of  98  hundredths,  i.e.  about 
2  hundredths,  i.e.  about  -02. 

This  is  useful  for  final  verification,  but  the  decimal  point 
is  best  fixed  by  the  rule  already  given.  If  the  simple  multi- 
plication and  division  are  accurately  performed,  the  correct 
fixing  of  the  decimal  point  is  a  simple  matter  to  even  an 
unintelligent  boy. 

Recurring  Decimals 

These  will  probably  rarely  be  used,  except  in  a  very 
simple  form.  Every  boy  ought,  however,  to  know  their 
significance,  though  as  a  subject  of  general  exposition  they 
are  now  generally  ignored,  perhaps  unwisely.  The  younger 
race  of  mathematicians  are  losing  familiarity  with  much 
that  is  interesting  in  the  theory  of  numbers.  Most  people 
know  of  course  that  \  -  -142857142857  .  .  .=  -14285?,  and 
that  if  we  multiply  this  group  of  6  figures  by  2,  3,  4,  5,  and 
6,  respectively,  we  obtain  products  giving  the  same  group 
of  figures  in  the  same  order,  each  succeeding  group  be- 
ginning with  the  next  higher  figure  of  the  group.  It  is, 
however,  less  commonly  known  nowadays  that  this  re- 
markable property  of  numbers  is  not  uniquely  characteristic 
of  the  sevenths  but  applies  to  all  prime  numbers  whatsoever, 
7  and  beyond,  and  that  the  grouping  within  the  groups  is 
sometimes  of  an  extraordinarily  interesting  character.  Teachers 
of  arithmetic  probably  lose  not  a  little  of  the  potential  effective- 
ness and  interest  in  this  subject  if  they  do  not  familiarize 
their  pupils  with  some  of  the  properties  of  numbers,  properties 
which  to  beginners  seem  almost  uncanny.  (See  Chapter  XLI.) 

As  to  circulating  decimals,  a  boy  should  be  taught  at 
least  this  much: 


DECIMAL  FRACTIONS  89 

Show  him  that  he  may  at  any  point  bring  to  an  end  the 
quotient  of  a  decimal  he  is  dividing,  showing  the  remainder 
as  a  vulgar  fraction. 

Thus  I  =  -14;  or  -1428  J  or  -142852. 

Reconvert,  say,  the  first:  -14=  -  ^  +  ^  =  1^1  =  '<><>  =  *. 

Then  the  boy  sees  that  the  scheme  is  justified. 

Again:  -J    =  «3333  .  .  .  apparently  without  end, 

so,  J    =  -1111  .  .  .  apparently  without  end, 

and,  J,J  —  -717171  .  .  .  apparently  without  end. 

But  we  can  bring  the  division  to  an  end  anywhere,  e.g.: 

i    =  -3333J,  or  -33J  or  -3£. 
1    =  -111?,  or-lj,. 
H  =-7171JJ  or-71JJ. 

Reconvert  these: 

ol 

.01    _   °  ;    __    1  O   __    :i  O   _   3 

;i  ""  io~"  ;l°  ~  '•»*-•»•- 

1  1 

.11       .   1<i  i    _    10   _    1 

'•'  ~      ~  lltf  ~  "' 


10 

.71?!     _      7      _i      x  i)  it    _      7        i        170      _    7100    _    71 

'  -1  '.»  (.)         10     '    TA         i  0     i"  u  y  0  o  —  i>  0  o  o         '.)  TT» 


171 

x  i)  it    _ 


Now  show  the  repeated  figures  in  a  decimal  division  this  way: 
•3333...   =   -3;     -1111...  =   -1;     -717171...  =   -tl. 

Thus  we  have  learnt  that 


Hence  to  convert  any  repeating  decimal  into  a  vulgar  fraction, 
we  make  a  denominator  of  9's,  viz.  as  many  as  there  are  places 
in  the  decimal. 

Thus  -Si  =  fJ;    '69  =       = 


90  CRAFTSMANSHIP  IN  MATHEMATICS 

If  there  are  non-repeating  figures  as  well  as  repeating  figures, 
e.g.  -5l6,  then 

1A      6-ia_6j!!-_  811. 
.516  =  _  -  _  -  ™, 

.01  A-  l'6  -    l"    -    i«    -    i 

016  -  loo  "  100  ~  TW  ~  **' 

The  commoner  forms  should  be  known,  especially  the  thirds, 
sixths,  and  twelfths: 


I  =  .16;     £  =  -83. 
^  =  -083;     ^  =  -416;     -/^  =  -583;     }4  =  -916. 

The  boys  should  know  that  when  the  denominator  of  a 
vulgar  fraction  contains  any  prime  numbers  except  2  and  5, 
the  conversion  of  the  fraction  to  a  decimal  is  bound  to  give 
a  circulating  decimal. 

Simplification  of  Vulgar  and  Decimal  Fractions 

There   are   certain   conventional   rules   about   signs;   for 
instance: 

1.  Multiplication  and  division  must  be  given  precedence 
over  addition  and  subtraction: 

3X  18 +15 -f-3  —  2       means       (3  X  18)  +  (15  ~  3)  -  2. 

2.  Multiplication  and  division  alone  must  be  worked  in  the 
natural  sequence  from  left  to  right: 

36  -r  9  X  2      means      (36  ~  9)  X  2. 

But  the  conventions  are  not  whole-heartedly  accepted;  they 
are  without  reason,  and  they  are  traps  for  the  unwary.  It  is 
unjust  for  examiners  to  assume  that  they  will  be  followed. 
The  above  examples  should  have  been  written  with  the 
brackets.  If  brackets  are  inserted  mistakes  need  not  arise. 

Here  is  a  complex  fraction  to  be  simplified,  taken  from 
one  of  the  best  textbooks  in  use.    Doubtless  the  question 


DECIMAL  FRACTIONS  91 

was  taken  from  an  examination  paper.     If  so,  the  examiner 
should  have  been  put  in  the  stocks. 

1-463       jj  of  141-75  -  j  X  88-125 
7-315          5  X  18-9  +  25  +  1-22 

If  given  at  all,  it  should  have  been  written 
^3  +  f  (lon^M^^ 

7-315        I      (5  X  18-9)  +  25  -f  1-22 


It  is  often  an  advantage  to  work  in  decimals  instead  of 
vulgar  fractions.     Example:    What  fraction  of  £21,  5$.  6d. 

.    -04255  X  -32    f  ri    „     0  ,  , 
1S  —  .  --  of  /I,  11$.  3rf.? 
•00016  *  ' 


„        .  -00016 

rraction  =  — 


•04255  x  -32    -,.  OK   ,.|f 
of  31-25  shill. 


425-5  shill. 
04255  x  -32  x  31-25 
^00016  X  425-5 

1          2 

x  &«  x  3125 


x 

1  1000 

=  6-25 


At  the  third  step,  both  numerator  and  denominator  were 
multiplied  by  109,  to  get  rid  of  the  decimals.  Boys  feel  more 
confidence  when  cancelling  whole  numbers.  But  in  A  Sets 
such  conversion  should  not  be  necessary. 

Decimalization  of  Money 

The  common  method  of  performing  arithmetical  opera- 
tions on  money,  weights,  and  measures  reduced  to  their  lowest 
denomination  has  the  advantage  of  simplicity  but  the  dis- 
advantage of  tediousness  and  cumbrousness.  It  is  certainly 
an  advantage  to  work  in  the  highest  denomination  when 
possible,  decimalizing  all  the  lower  denominations.  For 


92  CRAFTSMANSHIP   IN  MATHEMATICS 

instance,  if  we  have  to  multiply  £432,  175.  ±\d.  by  562,  it  is 
obviously  an  advantage  to  multiply  £432-86875  instead, 
provided  we  can  convert  into  the  decimal  form  at  once. 

But  the  rules  for  conversion  to  more  than  3  places  are  a 
little  too  difficult  for  slower  boys,  and  it  must  be  remembered 
that  if  multiplication  is  in  question  (and  this  is  often  the 
case),  exact  decimalization  is  necessary,  or  the  multiplied 
error  may  be  too  serious  to  be  negligible.  The  mil  invariably 
causes  trouble.  On  the  whole,  decimalization  methods  are 
advisable  in  A  Sets,  not  in  others. 

But  all  boys  should  be  taught  to  give  in  pounds  the 
approximately  equivalent  decimals  of  sums  of  money,  i.e. 
to  call  every  2/-,  •!;  every  odd  I/-,  -05;  every  -/6,  -025; 
every  farthing,  -001. 

Thus:    £3,   17s.   lOJrf. 

-  £3  +  16/-  +  I/-  +  -/G  +  IS/. 
=  £(3  -f-  -8  +  '05  -f  -025  -|-  -018) 
=  £3-893. 

Greater  certainty  and  greater  accuracy  is  obtained  by  the 
ordinary  method: 

\d.  -  -&d. 


=  -875.9. 

175.  lOJrf.    -  17-8755. 
=  £-89375 
£3,  17s.  10JJ.  -  £3-89375, 

the  boys  dividing  by  12  and  20  without  putting  down  the 
factors. 

The  converse  operation,  the  conversion  of  decimally 
expressed  money  into  pounds,  shillings,  and  pence,  is  most 
safely  and  quickly  performed  by  the  old-fashioned  multiplica- 
tion method  (by  20  and  by  12). 

Consider  £307-89275. 

£-89275  =  17-855s. 
•8555.  =  10-26rf. 
.'.  £307-89275  ==  £307,  175.  10  26rf. 

=  £307,  17s.  10^.  (+  -Old.). 


DECIMAL   FRACTIONS  93 

Numerous  tests  in  recent  years  have  shown  conclusively 
that  the  usual  decimalization  rules  are  productive  of  much 
inaccuracy  amongst  slower  boys.  But  decimalization  and 
reconversion  by  the  ordinary  methods  of  division  and  multi- 
plication are  easy  to  effect  and  are  often  advantageous  in 
practice. 

A  quick  boy  who  wanted  to  multiply  15s.  9rf.  by  2420 
would  probably  use  the  practice  method  (15s.  =  £  of  £1, 
9rf.  =  ~^(-  of  15s.,  and  he  would  see  at  once  that  the  product 
is  £1815  +  £90-75  or  £1905,  15s.),  and  he  would  not 
decimalize.  But  to  a  slow  boy  a  choice  of  methods  is  only 
an  embarrassment.  He  wants  one  method,  and  that  method 
without  frills  of  any  kind. 

Contracted  Methods 

Contracted  methods  of  multiplying  decimals  are  productive 
of  so  much  inaccuracy  that  their  use  with  average  boys  is  not 
advised.  In  A  Sets,  of  course;  in  B  Sets,  perhaps;  in  C  and 
D  Sets,  no;  though  in  A  and  B  Sets  logarithms  will  usually 
be  used  instead,  unless  the  sum  to  be  worked  is  so  simple 
that  ordinary  methods  are  quicker.  To  slower  boys  logarithms 
are  puzzling,  and  their  use  in  lower  Sets  is  not  recommended. 
But  no  boy  ought  to  be  allowed,  in  Forms  above  the  Fourth, 
to  show  in  his  working  the  figures  to  the  right  of  and  below 
the  heavy  line  in  a  sum  like  the  following. 

Divide  5-286143  by  37-29  (to  4  places). 

-1417 

3X^)5-286143 
3729 
1557  1 
1491  6 


65 
37 

82 


54 
29^ 
253 


26  103 
2  150 


94  CRAFTSMANSHIP  IN  MATHEMATICS 

I  have  found  that  even  slower  boys  soon  gain  confidence  in 
cutting  out  figure  after  figure  in  the  divisor,  instead  of 
bringing  down  figures  from  the  dividend.  The  doubtful 
"  carry  "  figures  worry  him  a  little  at  first,  but  not  for  long, 
and  he  soon  learns  to  understand  what  to  do  to  ensure 
accuracy  to  a  given  number  of  decimal  places. 

Never  encourage  average  boys  to  adopt  the  expert  mathe- 
matician's plan  of  multiplying  and  subtracting  at  the  same 
time  (Italian  method).  Boys  hate  it,  rarely  become  expert 
at  it,  and  make  mistakes  galore.  It  should  of  course  be  used 
by  boys  having  any  sort  of  real  mathematical  bent. 


CHAPTER  XI 

Powers   and   Roots.     The  A,   B,  C   of 
Logarithms 

Powers  and  Roots 

I  have  seen  four-  and  five-figure  logarithms  deftly  used  in 
Preparatory  Schools,  but  it  is  probably  not  wise  to  expect 
much  facility  before  the  age  of  14.  The  A,  B,  C  of  logarithms, 
as  a  simple  extension  of  work  on  powers  and  roots,  may, 
however,  readily  be  taught  a  little  sooner. 

At  first,  powers,  indices,  and  roots  should  always  be 
treated  arithmetically,  not  algebraically.  The  later  general- 
izations are  then  much  more  likely  to  be  understood. 

Some  typical  preliminary  exercises: 

(i)         54  =  5  X  5  X  5  X  5;    53  =  5  X  5  X  5; 
.-.   6*  X  63  =  (5  X  5  X  5  X  5)  X  (5  X  5  X  5) 

=  (5x5x5x5x5x6x5) 

=  5% 
/.  6«  x  5*  =  57. 


POWERS,  ROOTS,  LOGARITHMS  95 

Thus  lead  up  to  the  rule,  and  then  state  it  clearly,  that 
in  multiplication  of  this  kind  the  indices  are  added.  But 
impress  on  the  boys  that  the  operation  concerns  powers  of 
the  same  number  (in  this  case,  5),  though  any  number  may  be 
similarly  treated. 

(ii)       79  =  7x7x7x7x7x7x7x7x7; 

74  =  7x7x7x7; 


\X\X\X\ 


79 

L  =  7»-4  =  75 

74 


Now  lead  up  to  the  associated  rule  of  subtraction  of  indices, 
in  division  of  this  kind. 

(iii)          74  =  7X7X7X7; 

74_7x7x7x7__, 

"  74  ""  7~X  7  X  7~x~7  ~     ' 

.    74-4  ^  i     or     70  =  i       So  30  ^  1;   10o  =  j^ 

(iv)        (72)3  -  72  X  72  X  72 

-  (7  X  7  X  7  X  7  X  7  X  7)  =  76. 

Thus  lead  up  to  the  rule  as  to  multiplication  of  indices. 
State  categorically  that  such  results  always  hold  good,  and 
that  a  convenient  way  of  remembering  them  is  this: 


But  at  this  stage  do  not  talk  about  "  general  laws  ".  Let 
the  above  expressions  be  looked  on  merely  as  a  kind  of  short- 
hand for  collecting  up  several  results  actually  worked  out 
arithmetically. 

The  following  is  a  summary  of  a  particularly  effective 
first  lesson  I  once  heard,  given  to  a  class  of  boys  of  13,  on 
fractional  and  negative  indices. 

"  The  square  root  of  a  number  is  that  number  which 
when  squared  produces  the  original  number;  e.g.  the  square 
root  of  16  is  4;  of  81  is  9.  We  write  R16  =  4;  R81  =  9. 


96  CRAFTSMANSHIP  IN  MATHEMATICS 

(The  mathematician  writes  his  R  like  this:    '  v/>»  and  calls 
it  '  root  '.     Thus  *  V36  =  6  '  reads  '  root  36  is  6  '•) 
carefully    that    V6  X  V6  =  V^  =  V62  =  6;     that 
X  y'll  =  11;   and  so  on. 

"  The  cube  root  of  a  number  is  that  number  which  when 
cubed  produces  the  original  number.  We  show  the  operation 
by  writing  a  little  3  inside  the  <\A  Thus  4/125  =  4/5  X  5  X  5 
-  4/53  -  5;  4/1000  -  4/10  X  10  X  10~=  4/103  -  10; 
4/5  X  4/5  X  4/5  =  4/12.5  =  5.  _ 

"  And  so  on.    4/81  -  4/3~X  3x3x3  -  4/31  =  3. 

"  Now  suppose  the  index  is  a  fraction,  and  not  a  whole 
number.  What  does  5*  mean?  Well,  we  have  learnt  that 
52  x  52  =  52+2  =  54,  so  apparently  we  may  assume  that 
5*  X  5*  -  5*+*  =  51  -  5. 

"  But  V5  X  V5  =  5;   therefore  5*  =  V5- 
In  other  words,  5*  is  merely  another  way  of  writing  down  \/5. 
Similarly,    5*  X  5*  X  5*    -    5*  +  *f*    -    51   -   5;     therefore 
5*  =  4/5,  i.e.  5*  is  another  way  of  writing  down  4/5. 

"  Similarly,  4/7  -  71;    4/2  -  2*. 

"  What  does  8*  mean?  We  know  that  (34)3  =  312,  so 
apparently  we  may  assume  that  8*  =  (82)*,  i.e.  that  8J  means 
the  cube  root  of  82,  or  4/82,  or  4/64,  or  4. 

"  Similarly  53  =  4/53  -  4/125. 

"  Thus  we  have  learnt  that  the  numerator  of  a  fractional 
index  indicates  a  power  ',  and  that  the  denominator  indicates 
a  roo£. 

"  Again,  what  does  6~2  mean? 

"  Since  65  x  62  =  65+2  =  67,  apparently  we  may  assume 

that  65  X  6~2  -  6*-2  -  63. 

But  &  -f-  62  =  ^  -  63; 

.'.     6^  X  6~2  - 


or 


POWERS,  ROOTS,  LOGARITHMS  97 

Similarly,  5-3  =  -I  ;  also  1  =  7~4. 

Thus  we  may  conveniently  remember  that  a~n  and  —  are 
two  ways  of  writing  down  the  same  thing. 

"  Examples: 


7-i  _       _ 

75 


" 


A  root  form  in  a  denominator  is  often  troublesome, 
since  it  leads  to  difficult  arithmetic;  and  we  may  often  get 
rid  of  it  in  this  way: 

1=  J_  X  ^=^=  W6" 

A/5        V5        V5          5          " 

The  lesson  was  followed  up  by  a  few  very  easy  exercises. 
There  was  no  algebra,  save  the  "  shorthand  "  expressions 
utilized  as  mere  mnemonics.  The  teacher's  purpose  was  to 
make  the  boys  familiar  with  the  basic  facts  of  indices  (integral 
and  fractional,  +  and  —  ),  and  with  the  alternative  forms  of 
writing  down  the  same  thing.  Naturally  many  more  examples 
were  given  than  the  few  above  cited,  and  by  the  end  of  the 
lesson  the  boys  were  remarkably  accurate  in  their  answers  to 
"  mental  "  test  exercises  that  were  made  somewhat  severely 
searching. 

Boys  should  know  their  squares  up  to  202.  Extraction  of 
square  roots  may  be  taught  when  (a  +  b)2  is  known  in  algebra, 
though  boys  should  be  made  to  break  up  numbers  into 
factors  whenever  possible,  and  then  to  obtain  square  roots 
by  inspection.  Encourage  boys  to  leave  certain  types  of 
answers  in  surd  form,  but,  generally,  to  rationalize  their 

denominators;  thus  the  answer  —  -  would  not  be  acceptable, 

V7 

(E291)  8 


98 


CRAFTSMANSHIP  IN  MATHEMATICS 


but  y  5\/7  would.    All  boys  should  know  the  values  of 
V'S,  -\/5,  \/T,  to  2  places  of  decimals. 

The  Beginnings  of  Logarithms 

"  Mathematicians  were  long  ago  clever  enough  to  see 
how  they  could  use  indices  for  working  long  sums  in  multi- 
plication and  division.  Suppose  they  wanted  to  multiply 
together  two  large  numbers,  each  of  which  was  a  power  of  3. 
They  would  turn  to  a  book  of  *  tables  '  showing  the  powers 
of  3.  In  fact,  we  may  easily  make  up  a  little  table  for  ourselves: 
e.g.  3l  -  3;  32  =  9;  33  =  27;  34  -  81;  &c.  Here  is  a 
table  from  31  to  316.  In  the  first  column  we  write  the  index, 
in  the  second  the  corresponding  number. 


Index. 

Number. 

Index. 

Number. 

1 

1 

9 

19,683 

2 

9 

10 

59,049 

3 

27 

11 

177,147 

4 

81 

12 

531,441 

5 

243 

13 

1,594,323 

6 

729 

14 

4,782,969 

7 

2187 

15 

14,348,907 

8 

6561 

16 

43,046,721 

"  Now  for  some  exercises. 

1.  Multiply  19,683  by  729. 

Answer:  19,683  x  729  =  39  X  36  =  316  =  14,348,907. 


2.  Divide  43,046,721  by  531,441. 
Answer: 


531,441 


312 


3.  What  is  the  square  of  2187? 

Answer:  (2187)2  -  (37)2  =  314 


=  81. 


4,782,969. 


How  easy!  Instead  of  working  hard  sums,  we  simply  refer 
to  our  table,  and  add,  subtract,  or  multiply  little  numbers  like 
9,  6,  &c. 


POWERS,  ROOTS,  LOGARITHMS  99 

"  But  the  mathematician  would  set  out  the  first  sum  some- 
thing like  this: 

Index  of  the  answer  =  index  of  (19,683  x  729). 

=  index  of  19,683  +  index  of  729. 
=  9  +  6. 
=  15. 

"  But  in  the  table  the  number  corresponding  to  the  index 
15  is  14,348,907. 

.'.  Answer  =  14,348,907. 

Rather  a  roundabout  way,  isn't  it?  And  he  has  a  rather 
grand  word  which  he  prefers  to  the  word  index,  it  is  '  log- 
arithm '.  The  second  part  of  the  word,  -arithm,  you  already 
know;  the  first  part,  log  means  *  rule  '  or  '  plan  '.  Although 
logarithms  are  only  indices,  the  word  itself  suggests  a  clever 
"  arithmetical  plan  "  for  shortening  our  work,  and  you  must 
try  to  master  it. 

"  Our  little  table  contains  only  a  few  numbers,  and  there 
are  big  gaps  between  them;  e.g.  there  is  no  number  between 
27  and  81.  Now  27  -  33  and  81  =  34.  Would  it  be  possible 
to  obtain  a  number  between  27  and  81  by  finding  the  value 
of  33*?  Certainly  we  should  suspect  that  the  value  of  33* 
is  somewhere  between  27  and  81. 

"  We  know  that  33*  -  3*  -  ^  =  ^2187  =  46-8  (by  cal- 
culation). Hence  33*  does  lie  between  33  (—27)  and  34  (=81). 
Obviously  it  is  possible  to  put  into  our  table  as  many  fractional 
indices  as  we  like,  and  so  make  the  table  more  complete. 

"  The  3  which  we  have  made  the  base  of  our  calculations 
the  mathematician  calls  a  base.  Any  other  number  might 
be  used  instead,  and  in  point  of  fact  10  is  generally  used." 

The  boy  is  now  in  a  position  to  understand  that  (base)100 
=  natural  number,  and  he  may  be  introduced  to  a  short 
table  of  three-figure  logarithms,  a  table  that  may  be  included 
in  a  single  printed  page.  Give  a  variety  of  very  easy  examples, 
and  avoid  great  masses  of  figures.  It  is  enough  at  this  stage 
to  drive  home  the  main  principle.  There  is  much  to  be  said 


100  CRAFTSMANSHIP  IN  MATHEMATICS 

at  first  for  avoiding  the  word  logarithm  altogether,  and  for 
letting  the  boy  work  from  the  relation,  number  =  lOindMf. 
But  we  are  anticipating  Form  IV  work.  (See  Chap.  XVII.) 


CHAPTER  XII 

Ratio   and  Proportion 

Simple  Equations  Again 

If  4  chairs  cost  £20,  what  is  the  cost  of  15  chairs? 

4  chairs  cost  £20. 

/20 

.*.   1  chair  costs  — — . 
4 

••^    i    •  £20  X  15        --.., 

/.   15  chairs  cost  ^ —  £75. 

4 

This  method,  "  the  method  of  unity  ",  is  a  good  childish 
way  of  working  such  a  sum,  and  it  is  the  method  suitable 
for  boys  up  to  the  age  of  11.  At  about  this  age  the  notion  of 
ratio  should  be  introduced  and  it  should  gradually  supersede 
the  unitary  method. 

First,  revise  the  work  on  very  simple  equations. 

We  may  begin  a  sum  by  saying,  <k  What  number  of 
chairs  . . .  ?"  The  number  we  do  not  know;  we  have  to 
discover  it.  It  is  customary  to  let  the  symbol  x  represent  a 
number  not  yet  discovered,  and  to  argue  about  the  x  just  as 
we  argue  about  any  ordinary  number. 

How  many  chairs  can  I  buy  for  £45,  if  1  costs 

Let  the  number  of  chairs  be  x. 

Then  £5  x  x  =  £45, 
/.  5*  =  45. 
/.    *  =  Y  =  9- 

Similarly,  if  11  x  =  51,  x  =  ||  —  3. 


RATIO  AND  PROPORTION  101 

"  We  have  already  learnt  that  we  may  multiply  or  divide 
the  two  sides  of  an  equation  by  any  number  we  please, 
provided  that  we  treat  the  two  sides  alike;  e.g. 

If  x  =  12,  then  3x  =  36,   or   ~  =  —    and  so  on. 

If  we  have  an  equation  involving  fractions,  it  is  an  advantage 
to  get  rid  of  them  as  soon  as  we  can,  and  we  may  always  do 
this  by  multiplying  both  sides  of  the  equation  by  the  L.C.M. 

x       20 

of  the  denominators:    e.g.  let  the  equation  be  -  =  — .    The 

L.C.M.  of  9  and  12  is  36.  Multiplying  both  sides  by  36, 
we  have 

4x  --=  60. 
/.     x  =  15. 

Instead  of  using  the  L.C.M.  for  our  multiplier,  any  other 
C.M.  will  do,  though  this  will  mean  rather  harder  arithmetic. 
We  might,  for  instance,  use  the  product  of  the  denominators, 
viz.  108. 

x  =  20 

9        12* 
/.    12jc  =  9  X  20. 

x  —  15  (as  before). 

In  this  form  we  see  in  the  simplified  second  line  all  four 
terms  of  the  original  equation  (x,  9,  20,  12),  and  this  sim- 
plified second  line  might  have  been  obtained  at  once  from 
the  original  equation  by  cross  multiplying,  i.e.  by  mul- 
tiplying each  numerator  by  the  opposite  denominator. 
This  cross-multiplying  is  often  very  useful,  in  algebra  and 
geometry  as  well  as  in  arithmetic. 

"  From  cross-multiplying  it  follows  that  if  we  have  two 
equated  fractions,  a  numerator  and  the  opposite  denominator 
may  be  interchanged;  e.g.  £  =  \  -* ,  278  =  J.f ,  -^2~  =  -2\. 

"  Now  we  come  to  Ratio  and  Proportion." 


102  CRAFTSMANSHIP  IN  MATHEMATICS 

Ratio  and  Proportion 

If  1  sheep  cost  £3,  then, 


2  sheep  cost  £6 

3  „     „      £9 

4  „     „      £12 

5  „     „      £16 


7  sheep  cost  £21 
10  „  „  £30 
13  „  „  £39 
21  „  „  £63 


As  we  increase  the  number  of  sheep  we  increase  in  the  same 
proportion  the  number  of  pounds. 

Take  any  pair  of  numbers  (sheep)  from  the  first  column, 
and  the  corresponding  pair  of  numbers  (pounds)  from  the 
second,  say  the  last  but  two  and  the  last  in  each  case,  and 
convert  them  into  fractions,  thus: 


153* 


We  see  that  these  fractions  are  equal.  That  we  should 
expect,  for  10  bears  the  same  relation  to  21  as  30  bears 
to  63.  A  better  way  of  saying  it  is  that  the  ratio  of  10  to  21 
is  equal  to  the  ratio  of  30  to  63. 

"  We  know  that  the  sign  of  division  is  ~,  and  that  if  in 
the  place  of  the  two  dots  we  write  numbers,  e.g.  |,  we  have 
a  fraction,  and  that  the  fraction  means  5  divided  by  6.  Thus 
a  fraction  represents  a  quotient.  Similarly  a  ratio  represents 
a  quotient.  A  ratio  merely  shows  the  relation  between  two 
quantities,  viz.  how  many  times  one  is  contained  in  the  other. 
When  two  ratios  are  equal,  as  in  the  case  of  the  sheep  and 
pounds,  we  write  them  thus: 

10    30 

aT  —  *3> 

and  we  read, 

10  bears  the  same  ratio  to  21  as  30  bears  to  63. 

Such  a  statement  is  a  statement  in  proportion.  Sometimes 
we  read  '  10  is  to  21  as  30  is  to  63  ',  and  sometimes  *  10  over 
21  equals  30  over  63  '. 

"  Remember,  then:  a  statement  in  proportion  is  a 
statement  of  the  equality  of  two  ratios. 


RATIO  AND  PROPORTION  103 

How  many  pounds  of  tea  can  I  buy  for  40$.  if  6  Ib.  cost  15s.? 
Call  the  unknown  number  of  pounds,  x.    We  have  4:  terms 
viz.  2  lots  of  pounds,  2  lots  of  shillings.   Write: 

Lb.      Shillings. 
x  cost        40. 
6     „  15. 

Convert  each  pair  of  terms  into  a  ratio  or  fraction,  equate,  and 


6        15" 

.'.  5x  =  80. 
.'.     x  -  16. 

16  Ib.  of  tea  cost  40s.;  how  many  pounds  can  I  buy  for  155.? 

Lb.        Shillings. 
16  cost          40. 
x     „  15. 

«  16        40 

Equating,        —  =  —. 

x         15 

.-.  40#  =  240. 
/.       x  =  6. 

Find  the  cost  of  6  Ib.  of  tea  if  16  Ib.  cost  405. 

Lb.         Shillings. 
6  cost  x. 

16    „  40. 

Equating,  1  =  i. 

/.  2x  =  30. 
.-.    x  -  15. 

6  /6.  of  tea  cost  15$.;  etfAtf*  is  the  cost  of  16  Ib? 

Lb.         Shillings. 
6  cost  15. 

16    „  x. 

v  6        15 

Equating,  ---  =  —  . 

16       x 

:.  Qx  =  240. 


104  CRAFTSMANSHIP  IN  MATHEMATICS 

The  simple  scheme  applies  to  all  cases  of  direct  proportion 
write  down  the  4  terms  in  pairs;   equate;   solve  for  x. 

I  sell  a  horse  for  £47,  10$.,  thereby  losing  5  per  cent.    What 
should  I  have  sold  him  for  if  I  had  gained  5  per  cent? 

n  .       .    ^        ,  Representative 

Prices  tn  pounds  percentages. 

47J  95 

x  105 


47J        95 
Equating,         -«  = 


=  £52, 


95  2 

The  prices  of.  the  horse  are  in  direct  proportion  to  the  re- 
presentative percentage  numbers. 

Inverse  Proportion 

But,  of  course,  inverse  proportion  is  another  story.  In 
practice  it  is  relatively  rare,  and  is  thus  sometimes  overlooked. 

To  cover  a  floor  with  carpet  72  in.  wide  I  require  40 
yd.  from  the  roll;  if  the  carpet  is  only  half  the  width,  I  require 
twice  the  number  of  yards  from  the  roll;  if  only  one-third 
of  the  width,  then  3  times  the  number  of  yards.  We  may 
tabulate  thus: 

Running  yards.  Width  in  inches. 

40  72 

80  36 

160  18 

320  9 

Clearly  we  cannot  select  a  pair  of  terms  from  one  column  and 
equate  them  to  the  corresponding  pair  from  the  other.  One 
pair  has  to  be  inverted,  e.g. 


Thus  when  one  quantity  varies  inversely  as  another,  the  in- 


RATIO  AND  PROPORTION  105 

version  of  one  ratio  (it  matters  not  which)  is  necessary  before 
equating. 

Teach  the  boys  to  distinguish  between  direct  and  inverse 
proportion  by  asking  themselves  whether  when  one  quantity 
increases  the  other  increases  or  decreases,  and  to  distinguish 
them  on  paper  by  pointing  arrows  in  the  same  direction  to 
indicate  direct  proportion,  in  opposite  directions  to  indicate 
inverse  proportion. 

3  Ib.  of  tea  cost  8s.;   how  many  Ib.  will  cost  24$.? 

Lb.  Shillings. 

1  3  8  i 

*  x  24  * 

6  men  can  do  a  piece  of  work  in  20  days\  how  many  could 
do  it  in  15  days? 

Men.  Days. 

1  6  t  2° 

*  x  I    15 

Boyle's  Law  is  the  best  known  example  of  inverse  proportion 
in  science,  but  in  practical  life  examples  of  inverse  proportion 
are  much  less  common  than  those  of  direct,  and  the  conse- 
quence is  that  very  artificial  examples  are  often  invented  to 
illustrate  it.  "  Men  and  work  "  sums  are  often  silly.  "  If 
it  takes  20  men  to  build  a  house  in  20  days  ",  more  than 
one  maker  of  an  arithmetic  book  will  ask  us  to  believe  that, 
as  a  logical  consequence,  1000  men  could  build  the  house  in 
I  of  one  day. 

Never  use  the  old  form  of  proportional  statement,  :  ::  : . 
A  common  (and  meaningless)  form  of  statement  sometimes 
found  in  a  boy's  exercise  book  is 

36  :  40  ::  24. 

Always  let  the  equated  ratios  consist  of  two  fractions,  and 
make  the  boy  realize  that  the  particular  position  of  the  x 
(first,  second,  third,  or  fourth  place)  is  entirely  without 
significance. 


106  CRAFTSMANSHIP  IN  MATHEMATICS 


Examples  Acceptable  and   Unacceptable 

We  give  two  more  examples. 

1.  A  clock  which  was  1$  minutes  fast  at  10.45  p.m.  on 
2nd  December  was  8  minutes  slow  at  9  a.m.  on  1th  December. 
When  was  it  exactly  right? 

This  problem,  like  most  other  problems,  requires  a 
preliminary  discussion.  By  judicious  questioning,  help  the 
boys  to  cast  it  in  a  simpler  form: 

A  slow-going  clock  loses  9^  of  its  false  minutes  in  106J  true 
hours.  In  how  many  hours  will  it  lose  1$  of  its  false  minutes? 

"  Minutes  "  lost  by  slow  clock     during    Hours  of  true  clock 

;  ;•  106i  i 

i-J-  x 

9J    ^    lOGJ 

ij         *T* 

85  _   425 
"      13   "    4*"* 

JL  =  _^ 

"     13  ~  4i" 
/.      x  =   161, 

i.e.  the  slow  clock  was  right  16J  hours  after  10.45  p.m.  on 
2nd  December,  i.e.  at  3  p.m.  on  3rd  December. 

2.  It  takes  8  men  6  days  to  mow  a  field  of  grass.     How 
long  would  it  take  20  men  to  do  it? 

Days  Men 

f 
• 


8 
20 


6  =   20 

x  ~~    8" 

x  =  2f  . 


But  although  2|  days  is  the  orthodox  answer,  the  time  would 
really  be  rather  less.    Men  mowing  a  field  for  6  days  would 


RATIO  AND  PROPORTION  107 

find,  in  the  growing  season,  the  work  much  harder  on  for 
example,  the  sixth  day  than  on  the  first,  so  that  the  amount 
of  grass  cut  would  not  be  equally  distributed  over  the  6  days. 
The  answer  as  calculated  is  but  a  rough  approximation. 
Writers  of  textbooks,  and  some  examiners,  are  so  often  out 
of  touch  with  practical  life  that  it  may  be  useful  to  append 
a  few  absurd  questions  of  the  type  supposed  to  be  examples 
of  Ratio  and  Proportion: 

1.  It  takes  3  minutes  to  boil  5  eggs.    How  long  would  it 
take  to  boil  6  eggs? 

2.  A  man  rides  a  bicycle  at  the  rate  of  20  miles  an  hour. 
How  far  could  he  travel  in  92|  hours? 

3.  My  salary  is  £500  a  year  and  I  save  £50  a  year.    How 
long  shall  I  take  to  save  £10,000? 

4.  My  brother  weighed  24  Ib.  when  he  was  3  years  old. 
How  much  will  he  weigh  when  he  is  45  years  old? 

5.  A  rope  stretches  |  in.  when  loaded  with  1  cwt.    How 
much  will  it  stretch  when  loaded  with  10  tons? 

6.  It  cost  £1  to  dig  and  line  a  well  2  ft.  deep.     How 
much  will  it  cost  to  dig  and  line  a  well  100  ft.  deep? 

7.  A  stone   dropped   down   an  empty  well   16   ft.  deep 
reaches  the  bottom  in  1  second.  What  is  the  depth  of  another 
well,  if  a  stone  takes  5  seconds  to  reach  the  bottom? 

Another  point:  if  the  answer  to  problems  concerning  men 
and  work  comes  out  to,  say  4J  men;  instruct  the  boy  to  say 
5  men,  with  an  explanatory  note. 

Until  a  boy  is  thoroughly  well  grounded  in  Ratio  and 
Proportion,  the  formal  statement  of  the  ratios  is  desirable. 
But  at  least  the  abler  boys  in  the  top  Forms  may  be  allowed 
to  do  as  mathematicians  themselves  do — write  down  the 
odd  term  and  multiply  at  once  by  the  fraction  determined 
by  the  ratio  of  the  other  two  terms. 

3 1  Ib.  of  tea  cost  llf  shillings;  find  the  cost  of  1\  Ib. 

Cost  =  llfs.  x  -f . 
3s 


108  CRAFTSMANSHIP  IN  MATHEMATICS 

"  Compound  "  Proportion 

"  Double"  or   "  Compound  "   "Rule  of  Three  ". 

For  the  most  part  the  typical  sums  given  by  the  textbooks 
to  illustrate  this  "  rule  "  have  little  relation  to  practical  life. 
Occasionally  they  are  legitimate  enough,  and  then  they  may 
be  regarded  as  just  a  simple  extension  of  the  simpler  two- 
ratio  examples  already  considered.  The  terms  may  be 
arranged  in  their  natural  pairs,  converted  into  ratios,  these 
marked  (with  arrows)  direct  or  inverse,  then  multiplied  out. 
If  16  cwt.  are  carried  63  miles  for  £6,  6$.,  what  weight 
can  be  carried  112  miles  for  £2,  16s.? 

cost  in 
cwt.  miles.  shillings 

\      x  A   112  I      56 

*   16  I      63  *   126 


x 

112        126 


=  4. 


Here  is  another,  one  of  the  commoner  types,  taken  from  one 
of  the  best  of  the  textbooks:  If  36  men  working  8  hours  a 
day  for  16  days  can  dig  a  trench  72  yd.  long,  18  ft.  wide,  12/J. 
deep,  in  how  many  days  can  32  men  working  12  hours  a  day 
dig  a  trench  64  yd.  long,  27  ft.  wide,  and  18  ft.  deep? 

The  example  is  not  practicable.  Men  working  12  hours 
a  day  can  not  do  1-J  times  as  much  work  as  men  working 
8  hours  a  day.  The  cost  of  digging  a  trench  18  ft.  deep  is 
more  than  \\  times  the  cost  of  digging  one  12  ft.  deep.  The 
deeper  the  trench  the  more  expensive  it  is  to  get  out  the 
excavated  earth.  The  cost  does  not  necessarily  vary  as  the 
width  of  the  trench;  if  timbering  the  sidfcs  is  necessary  (a 
serious  additional  item  of  expenditure),  a  little  extra  width 
would  not  add  appreciably  to  the  cost.  But  more  than  this: 
for  excavation  work,  steam  navvies  have  largely  replaced 
manual  labour. 

So  it  is  with  a  large  number  of  the  textbook  exercises: 


RATIO  AND  PROPORTION  109 

they  have  no  relation  to  practical  life.  Here  is  one  more, 
from  a  really  excellent  textbook. — If  10  cannon  which  fire 
3  rounds  in  5  minutes  kill  270  men  in  1|  hours,  how  many  cannon 
which  fire  5  rounds  in  6  minutes  will  kill  500  men  in  1  hour? 
Did  the  man  who  made  up  this  problem  claim  to  be  a  mathe- 
matician, or  a  soldier,  or  a  humorist?  It  is  a  shocking  thing 
that  school  boys  are  made  to  waste  their  time  over  the  pretence 
of  "  solving  "  problems  of  this  kind. 


CHAPTER  XIII 

Commercial  Arithmetic 

No  branch  of  arithmetic  is  more  important,  and  yet  it 
need  not  take  up  a  very  great  deal  of  time.  For  the  most 
part,  the  work  consists  of  the  application  of  principles,  already 
learnt,  to  business  relations  in  practical  life.  Once  the  boy 
grasps  the  inner  nature  of  the  business  relation,  the  arith- 
metic should  give  him  little  trouble.  But  "  hard  "  sums, 
especially  sums  involving  great  masses  of  figures,  are  rarely 
if  ever  necessary.  Give  ample  practice  in  working  easy 
exercises  and  so  make  the  boy  thoroughly  familiar  with  the 
A  B  C  of  commercial  life. 

Percentages 

Teach  the  meaning  of  "  per  cent  "  thoroughly.  We 
require  a  numerical  standard  of  reference  of  some  kind,  and 
the  number  100  has  been  accepted  as  the  most  convenient, 
though  any  other  would  do  instead.  It  is  a  disadvantage 
that  100  is  not  divisible  by  3. 

5  per  cent  means  5  per  100  or  1  jfo.  Drive  this  cardinal 
fact  well  home:  everything  hangs  upon  it.  5  per  cent  of 


110  CRAFTSMANSHIP  IN  MATHEMATICS 


£1  =  y^  of  20*.  =  1*.;  2J  per  cent  of  £1  =  6df.  Let  these 
two  results  be  the  pegs  of  plenty  of  mental  arithmetic;  e.g. 
1\  per  cent  of  £1  =  3  times  6rf.  =  Is.  6d.,  and  7£  per  cent 
of  £20  =  1$.  6d.  X  20  =  30^.  ;  and  so  on. 

Representative  percentage  numbers  (as  they  are  usefully 
called)  is  the  next  thing  to  drive  home.  When  we  buy  a 
thing,  it  may  be  assumed  that  we  buy  it  at  the  standard 
price  which  is  represented  by  100.  If  we  sell  the  thing  at 
10  per  cent  profit,  we  sell  it  at  a  price  represented  by  110; 
if  we  sell  it  at  a  loss  of  15  per  cent,  we  sell  it  at  a  price 
represented  by  the  number  85.  This  notion  is  of  fundamental 
importance.  The  majority  of  exercises  grouped  under  the 
term  "  percentages  "  or  "  profit  and  loss  "  are  cases  of 
simple  proportion,  the  two  terms  of  one  ratio  consisting  of 
money  and  the  two  terms  of  the  other  ratio  consisting  of 
representative  percentage  numbers. 

How  much  is  12f  per  cent  of  £566,  13*.  4rf.? 


Direct  proportion  example  (common):  If  a  debt  after  a 
deduction  of  3  per  cent  becomes  £210,  3s.  4rf.,  what  would  it 
have  become  if  a  deduction  of  4  per  cent  had  been  made? 

Reduced  debts. 

97  ,    £210* 

96  *       x 

13        16 


i     i 

=  £208. 

Inverse  proportion  example  (comparatively  rare):  A 
fruiterer  buys  shilling  baskets  of  cherries,  30  in  a  basket.  He 
also  sells  them  at  a  shilling  a  basket^  but  24  in  a  basket.  What 
profit  per  cent  does  he  make? 


COMMERCIAL  ARITHMETIC  111 

The  smaller  the  number  of  cherries  he  sells  in  a  basket, 
the  larger  his  profit.    Hence  the  proportion  is  inverse. 


Cherries  per 
Basket. 

Representative 
%  Nos. 

1    30 
*  24 

A  100 

'       X 

x  = 

»«•"£ 

=  125. 

This  representative  percentage  number  shows  that  the  profit 
is  25  per  cent.  (Strictly,  the  answer  is  not  right,  as  no  allow- 
ance is  made  for  the  necessary  purchase  of  new  baskets.) 

Another  inverse  proportion  example:  If  eggs  are  bought 
at  21  for  Is.,  how  many  must  be  sold  for  a  guinea,  to  give  a 
profit  of  12|  per  cent? 

The  selling  price  is  represented  by  112|,  a  number 
greater  than  100;  the  number  of  eggs  sold  for  a  guinea  must 
be  smaller  than  the  number  bought  for  a  guinea.  Hence 
the  proportion  is  inverse. 

Ar     r     01  /  Representative 

No.  for  21/- 


441  I    100 

x  *   112J 

.-  441  X^ 
112J 

=441  Xf 
=  392. 


Simple  Interest 


The  kind  of  examples  really  necessary  should  cause 
little  trouble.  Even  a  slow  boy  readily  understands  the  main 
principles.  As  soon  as  he  has  learnt  what  5  per  cent  per 
annum  means,  he  can  follow  this  reasoning: 


112  CRAFTSMANSHIP  IN  MATHEMATICS 

Interest  on  £100  at  5  per  cent  per  annum  for  1  year 


Interest  on  seven  times  £100,  i.e.  on  £700  for  1  year 

=  £700  X  Ttld- 
Interest  on  £700  for  3  years  =  £700  X  T|o  X  3. 

There  is  now  an  excellent  opportunity  for  establishing  a 
simple  algebraic  formula: 

Let  I  =  Interest. 
„  P  =  Principal. 
„  R  =  Rate  per  cent. 
,,  T  =  Time  in  years. 

Then          I-PxTJ*R^PTR 
men          i  1Q()  -  10Q. 

The  technical  term  "  amount  "  should  also  be  explained: 
A-P  +  I. 

As  interest  is  usually  paid  half-yearly,  "  5  per  cent  per 
annum  "  (as  in  the  case  of  Government  Stock)  generally 
represents  rather  more  than  its  nominal  value.  This  should 
be  explained. 

The  use  of  the  formula  is  quite  legitimate,  provided  the 
boy  has  learnt  to  establish  it  from  first  principles;  and  equally 
he  may  be  allowed  to  deduce  the  subsidiary  formulae,  arguing 
in  this  way: 

Since  from  first  principles 
i  _  PTR 

loo' 

/.  I  X  100  =  PTR. 

P  =  L2L1?0.  T  -  L*  -I??-  R  =  *  x  1QO 

"       ~      TR     '  PR     ;  PT     " 

But  in  practical  life  these  subsidiary  problems  (to  find  P  or 
T  or  R)  are  very  rarely  wanted,  and  it  is  not  worth  while 
to  let  boys  waste  time  over  working  a  large  number.  An 
occasional  example,  mainly  to  give  facility  in  the  use  of  the 
formula,  is  enough. 


COMMERCIAL  ARITHMETIC  113 


Compound  Interest 

It  is  enough  to  tell  a  boy  to  find  out  what  will  be  due  to 
him  if  he  places  in  the  Bank  £100  on  deposit  and  allows 
it  to  remain  there  for  2  or  3  years,  the  interest,  say  at  "  4 
per  cent  ",  being  undrawn.  Two  minutes  of  explanation 
will  show  him  how  to  work  the  sum,  each  half-year's  interest 
being  added  to  the  Principal  as  it  becomes  due.  A  little 
later  on,  instruction  will  be  necessary  as  to  shorter  procedure 
in  calculation,  but  to  give  up  time  to  the  working  of  numerous 
examples  is  inadvisable.  Bankers  never  work  compound 
interest  sums:  they  merely  refer  to  ready-made  tables, 
prepared  by  mathematical  hacks  for  all  the  world  to  use. 
Do  not  let  the  boys  waste  time  over  such  useless  work, 
especially  as  the  time  is  so  badly  wanted  for  other  things. 
On  the  other  hand,  see  that  they  really  do  understand  main 
principles,  and  can  readily  apply  them  to  simple  cases. 

The  subject  may,  of  course,  be  resumed  in  the  Fifth  or 
Sixth  Form,  should  the  general  mathematical  theory  of 
interest  and  annuities  be  taken  up. 


Present  Worth  and  Discount 

Here  again  the  principles  are  important  and  are  very 
easily  mastered.  Their  use  may  be  amply  illustrated  by 
reference  to  a  few  easy  examples.  Do  not  forget  to  give  a 
clear  explanation  of  Bills  of  Exchange  and  Promissory 
Notes. 

The  boy  already  knows  from  his  interest  sums  that 

Amount  =?  Principal  +  Interest. 

In  Discount  sums,  three  new  terms  are  used,  and  really  they 
are  identical  with  the  three  just  mentioned: 

Sum  Due  =  Present  Worth  +  Discount. 

(1291)  9 


114 


CRAFTSMANSHIP  IN  MATHEMATICS 


Here  are  two  exactly  analogous  examples  in  direct  proportion 
of  the  normal  type. 

What  is  the  Principal  that 
will  produce  an  Amount  of  £840 
in  3  years  at  4  per  cent? 

When  we  have  found  the 
Principal,  we  can  subtract  it  from 
the  Amount,  and  so  obtain  the 
Interest. 


Principal, 
x 


Amount. 
£840 


What  is  the  Present  Value  of 
£840,  the  Sum  Due  at  the  end 
of  3  years,  the  interest  being  4 
per  cent? 

When  we  have  found  the 
Present  Value,  we  can  subtract 
it  from  the  Sum  Due,  and  so 
obtain  the  Discount. 


Present  Value. 

x 


Sum  Due. 
£840 


But  there  is  only  one  term  for  each  of  the  Ratios.  Where  are 
the  others?  We  have  to  invent  them. 

We  do  not  know  the  value  of  x,  the  Principal  that  will 
amount  to  £840  in  3  years  at  4  per  cent. 

We  do  not  know  the  value  of  x,  the  Present  Value  of  the 
sum  £840  due  in  3  years  at  4  per  cent. 

But  we  may  take  any  sum  we  please  and  invest  it  for  3  years 
at  4  per  cent.  £100  is  as  good  a  sum  as  any. 

£100  invested  for  3  years  at  4  per  cent  yields  £12  interest. 

Thus  £100  is  the  Principal  that  Amounts  to  £112  in 
3  years  at  4  per  cent,  and  £100  is  the  Present  Value  of  £112, 
the  sum  due  in  3  years  at  4  per  cent.  Now  we  may  complete 
our  Ratios. 


Principals.         Amounts. 
*   100  ^      112 


x 

ioo 


840 
If2' 


15 


=  £750  (Principal) . 

Interest  (if  required) 
-  £840  -  £750  =  £90. 


Sums  Due. 
I    £840 
*      112 


Present  Values. 

4x 
100 

x    __  840 
100       112* 

15 

/.  £*  =  100  x  |g 

2 
=  £750  (Present  Value). 

Discount  (if  required) 
=  £840  -  £750  =  £90. 


COMMERCIAL  ARITHMETIC  115 

Remind  the  boy  that  this  True  discount  is  never  heard  of 
in  practice.  The  Bill  Broker's  discount,  which  he  deducts, 
is  really  the  interest  on  the  whole  Sum  Due.  It  is  exactly  the 
same  as  calculating  Interest  on  the  Amount  instead  of  on  the 
Principal,  a  thing  the  banker  would  (naturally)  never  dream 
of  doing.  Let  the  boy  compare  the  two  things,  and  see  for 
himself  that  when  the  Banker  deducts  interest  on  the  Sum 
Due  instead  of  on  the  Present  Value,  the  customer  receives, 
as  Present  Value,  a  sum  rather  less  than  by  arithmetic  he 
is  entitled  to. 

Exercises  in  Present  Value  and  Discount  are  hardly 
worth  doing,  unless  they  are  very  simple  and  can  be  done 
quickly. 

Stocks  and  Shares 

Nothing  is  more  important  in  arithmetic  than  a  working 
knowledge  of  stocks  and  shares  and  of  financial  operations. 
Whatever  views  political  extremists  may  take  of  a  roseate 
financial  future,  we  have  to  deal  with  the  hard  facts  of  the 
present  day,  when  it  behoves  every  member  of  society  to 
save,  and  to  invest  his  savings. 

But  do  not  make  boys  waste  their  time  by  working  through 
the  useless  examples  on  stocks  and  shares  given  in  many  of 
the  older  textbooks. 

The  first  stile  for  the  boy  to  get  over  is  the  distinction 
between  stock  and  money,  and  there  is  no  better  way  than 
to  turn  the  whole  class  into  an  imaginary  Limited  Liability 
Company  with  its  own  Directors.  To  play  a  game  of  this 
kind  is  worth  while.  Let  the  Directors  draw  up  a  simple 
Prospectus  and  invite  subscriptions  at  par.  A  week  later 
let  the  Directors  report  some  disaster — perhaps  the  destruc- 
tion of  property  by  fire — and  an  inevitable  fall  in  the  expected 
interest.  Some  shareholders  will  become  anxious  and  will 
be  willing  to  sell  at  90  or  even  lower.  And  so  on.  A  little 
reality  of  this  kind  is  worth  ten  times  the  value  of  a  long 
sermon  on  the  subject.  If  a  boy  pays  £100  (any  sort  of  paper 


116  CRAFTSMANSHIP  IN  MATHEMATICS 

token  will  do)  and  receives  a  Certificate  for  100  £1  shares, 
and  then  has  to  part  with  his  shares  at,  say,  18$.  each,  a 
sense  of  reality  is  brought  home  to  him.  Perplexity  about 
stocks  and  shares  is  almost  always  due  to  a  hazy  understanding 
of  the  reality  which  underlies  it  all.  As  always,  the  trouble 
is  with  the  slower  boys.  The  quicker  boys  pick  up  the  threads 
readily  enough. 

There  are  numerous  facts  for  the  boys  to  understand  and 
remember,  as  well  as  sums  to  work.  Explain  the  nature  of 
debentures,  preference  shares,  the  different  kinds  of  ordinary 
shares,  their  relative  value  and  relative  safety.  Warn  the 
boys  never  to  invest  without  taking  advice,  and  never  in  any 
circumstances  to  invest  in  a  new  flotation.  Explain  "  gilt- 
edged  "  securities,  and  point  out  the  relative  safety  of  Govern- 
ment stock,  though  even  this  may  fall  seriously  in  value 
(compare  the  present  price  of  Consols  with  the  price  fifty 
years  ago).  Insist  that  a  large  interest  connotes  a  big  risk, 
that  financial  greed  spells  disaster.  Impress  upon  the  boys 
that  the  financial  world  is  full  of  sharks. 

The  old  days  of  a  brokerage  of  ^  per  cent  have  passed 
away,  and  thus  many  of  the  sums  in  the  older  text- 
books are  out  of  date.  Stockbrokers'  charges  now  include 
Government  Stamp  Duty,  Company's  Registration  fee, 
and  Contract  Stamp.  Give  the  class  a  short  table  of 
charges  to  be  entered  in  their  notebooks,  for  permanent 
reference,  e.g. 

Purchases  £50+  to  £75,  total  charges  18s.  3d. 
Purchases  £75+  to  £100,  total  charges  £1,  3s.  3rf., 
and  so  on. 

All  ordinary  "  examples  "  in  stocks  and  shares  are  in- 
stances of  simple  proportion  (nearly  always  direct:  there  is 
little  point  in  puzzling  young  boys  with  the  rule  "  the  amount 
of  stock  held  is  inversely  proportional  to  the  price  "),  and 
they  call  for  no  comment. 

Examples  on  the  purchase  and  on  the  sale  of  small  amounts 
of  stock  and  small  numbers  of  shares  are  the  only  exercises 


COMMERCIAL  ARITHMETIC  117 

that  need  be  given.     Let  the  exercises  be  typical  of  those 
that  in  practical  life  the  average  man  engages  in. 


Other  Commercial  Work 

Rates  and  Taxes. — A  simple  explanation  of  and  a  variety 
of  exercises  in  these  are  of  great  importance.  Explain  the 
increase  in  both  rates  and  taxes  during  the  last  few  years. 
Distinguish  carefully  between  expenditure  by  the  Govern- 
ment and  expenditure  by  Local  Authorities,  and  show  why 
both  kinds  of  expenditure  are  inevitable.  Explain  how  taxes 
are  imposed  and  how  rates  are  levied.  Let  exercises  be  easy, 
but  devise  them  to  illustrate  principles  and  to  give  an  inner 
meaning  to  things.  "  Rateable  values  "  is  another  thing  to 
be  explained. 

And  there  are  numerous  other  things,  of  which  it  behoves 
every  intelligent  person  to  have  at  least  an  elementary  know- 
ledge, things  which  only  a  mathematical  teacher  can  handle 
effectively.  We  mention  a  few:  Income  Tax  and  its  assess- 
ment, its  schedules,  its  forms  and  the  correct  method  of 
filling  them  up;  rent,  house  purchase,  mortgages;  the  raising 
of  loans  by  public  bodies  and  by  private  persons;  insurance 
of  all  kinds,  especially  life  insurance,  Health  and  Unemploy- 
ment insurance;  policies  (especially  "  all-in  "  policies)  and 
premiums;  pensions,  annuities,  the  keeping  of  personal 
accounts,  thrift,  household  economics;  banks  and  saving; 
the  Post  Office  bank  and  National  Saving  Certificates;  co- 
operative stores  and  their  financial  basis;  building  societies; 
insurance  tables  and  how  to  read  them  (a  Sixth  Form  ought 
to  have  some  knowledge  of  their  actuarial  basis).  There 
are  numerous  tables  of  very  useful  kinds  in  Whitaker  that 
every  boy  ought  to  be  made  to  understand,  and  by  means  of 
them  an  arithmetic  teacher  may  devise  exercises  of  a  very 
valuable  kind. 

A  particularly  useful  syllabus  on  the  arithmetic  of 
citizenship  is  given  in  the  appendix  of  the  1928  Report  of 


118  CRAFTSMANSHIP  IN  MATHEMATICS 

the    Girls'    Schools    Sub-committee    of   the    Mathematical 
Association. 

Books  on  arithmetic  to  consult: 

1.  The  Psychology  of  Arithmetic,  Thorndike. 

2.  Lecons  d'Arithme'tique,  Tannery  (Armand  Colin). 

3.  The  Teaching  of  the  Essentials  of  Arithmetic,  Ballard. 

4.  The  Tutorial  Arithmetic,  Workman. 

5.  The  Groundwork  of  Arithmetic,  Punnett. 

6.  The  Small  Investor,  Parkinson. 


CHAPTER  XIV 

Mensuration 

Simple  Formulae 

Easy  problems  involving  actual  measurements  will  be 
embodied  in  the  mathematical  course  for  children  below  the 
age  of  11,  by  which  time^  boy^ught  to  be  familiar  with  the 
mensuration  of  rectangular  areas  and  rectangular  solids  and 
to  be  able  to  work  easy  conversion  (reduction)  sums  in  linear, 
square,  and  cubic  measures.  He  ought  also  to  have  learnt 
to  measure  up  the  area  of  the  classroom  floor  and  walls,  and 
to  express  his  results  in  formula  fashion)  e.g.  area  of  floor 
=  /  X  b\  area  of  the  four  walls  =  2(1  +  b)h 

\He  should  now  be  taught,  if  he  has  not  been  taught 
before,  to  make  paper  models  of  cubes  and  cuboids,  and 
from  a  consideration  of  the  "  developed  "  surfaces  of  these, 
laid  out  in  the  form  of  "  nets  ",  to  devise  formulae  for  cal- 
culating the  areas;)  e.g.  of  a  cube,  6/2;  of  a  square  prism, 
4/z/  +  2a2  or  2a(2l  +  a)\  of  a  brick,  2(lb  +  It  +  bt).  £The 
memorizing  of  these  formulae  is  not  worth  while,  but  they 
are  worth  working  out  as  generalizations  from  particular 
examples;  and  when,  once  more,  numerical  values  are 
assigned  to  them,  it  makes  early  algebra  very  real.j) 


MENSURATION  119 

The  Papering  of  Rooms 

Some  attention  must  be  given  to  the  stock  problems 
on  the  papering  of  rooms,  but  it  is  not  worth  while  to  take 
time  over  measuring  up  doors,  windows,  and  fire-places; 
assume  that  the  walls  are  unbroken  planes,  and  the  room 
rectangular.  Nor  is  it  worth  while  to  divide  the  perimeter 
of  the  room  by  21  in.  to  find  the  necessary  number  of  strips 
of  paper.  There  is  bound  to  be  a  good  deal  of  paper  wasted, 
especially  if  the  pattern  is  elaborate.  Hence  it  is  enough  to 
take  the  total  wall  area  2(1  ~\~  b)h,  divide  this  by  the  area  of 
one  roll  of  paper,  36  ft.  X  If  ft.  or  63  sq.  ft.  or  9  sq.  yd., 
and  so  obtain  the  necessary  number  of  rolls.  If  the  answer 
comes  out  to  13£  rolls,  evidently  14  are  wanted,  perhaps 
15  because  of  waste;  perhaps  13  or  even  12  would  do,  because 
of  windows,  doors,  &c.  A  paper-hanger  never  measures  up 
a  room  with  any  degree  of  accuracy;  his  estimate  is  very 
rough  and  always  done  by  rule  of  thumb.  There  is  really 
no  point  in  giving  boys  such  problems  to  work,  especially 
when  it  is  remembered  what  a  large  number  of  problems, 
depending  on  accurate  measurements,  may  be  culled  from 
the  boys'  physics  course. 

The  Carpeting  of  Floors 

The  carpeting  of  floors  is  generally  considered  to  give 
an  easier  type  of  problem  than  the  papering  of  walls,  but 
the  problem  in  practice  is  a  little  tricky.  If  from  an  ordinary 
27-in.  wide  roll  a  carpet  has  to  be  made  up  to  fit  the  usual 
rectangular  room,  it  is  unlikely  that  the  width  of  the  room 
is  an  exact  multiple  of  27  in.,  in  which  case  the  last  of  the 
strips  cut  off  the  roll  will  be  too  wide,  and  there  will  be 
waste;  and  yet  the  whole  of  that  strip  will  have  to  be  pur- 
chased, as  the  pattern  cannot  be  "  matched  ".  If  the  carpet 
is  plain,  and  the  purchaser  does  not  object  to  patching, 
then  the  exact  amount  required  may  be  cut  from  the  roll, 


120  CRAFTSMANSHIP  IN  MATHEMATICS 

though  the  vendor  might  not  agree  to  cut  to  the  small  fraction 
of  a  yard. 

Consider  a  floor  18'  X  12',  and  carpet  2'  3"  wide. 

1.  Let  the  carpet  be  plain  (patternless).     Area  of  floor 
=  24  sq.  yd.     Required  number  of  running  yards  from  the 
roll  —  24  -r  f  =  32.      This  will  give   5  strips,   each   6  yd. 
long,  and  2  running  yards  (a  piece  6'  X  2'  3")  over.     This 
strip  of  6'  X  2'  3"  will  have  to  be  cut  up  to  cover  a  space 
18'  by  I)",  so  that  it  will  be  cut  into  3  pieces  each  6'  long, 
placed  end  to  end,  the  width  of  these  being  9". 

2.  Let  the  carpet  show  a  design,  the  width  being  the  same 
as  before.     Evidently  at  least  6  strips,  each  6  yd.  long,  or 
36  yd.  in  all,  must  be  cut  from  the  roll.     It  is  highly  im- 
probable that,  if  the  strips  are  cut  to  exact  length,  they  would 
match  when  laid  side  by  side.    There  would  be  a  good  deal 
of  waste,  depending  on  the  size  of  the  design.    The  problem 
cannot  be  brought  within  the  scope  of  classroom  arithmetic: 
all  the  factors  are  not  available. 

3.  A    more    practical    problem   for   the    classroom   is   to 
estimate  the  amount  of  plain  carpet  required  to  cover  a  room 
of  given  size  with  a  minimum  number  of  complete  strips, 
allowing  the  surplus  width  to  determine  an  equal  all-round 
border  (to  be  stained  or  covered  with  linoleum).   For  instance, 
the  5  strips  above  mentioned  would  leave  a  surplus  width  of  9". 
If  the  strips  are  placed  together  centrally,  there  will  be  a  width 
of  4|"  to  spare  at  each  end  of  the  room.     Hence  we  must 
arrange  for  a  complete  border  of  4|-"  all  round  the  room.   Thus 
the  5  strips  will  not  now  be  18'  long,  but  17'  3"  long,  and  the 
amount  to  be  cut  from  the  roll  will  be  17'  3"  by  5  (28|  yd.). 

Thus  the  area  of  the  room  =  18'  X  12';  of  the  carpet, 
17'  3"  X  11'  3";  and  the  required  number  of  running  yards 
from  the  2'  3"  wide  roll  -  28|. 

,  If  the  whole  floor  had  been  covered,  and  patching  was 
allowable,  the  number  of  running  yards  required  =  32;  if 
patching  was  not  allowable,  the  number  of  running  yards  =  36 
(leaving  a  waste  piece  6  yd.  long  and  18"  wide,  with  an  area 
of  3  sq.  yd.,  equal  to  4  running  yards). 


MENSURATION  121 

Does  it  not  all  come  round  to  this — that  these  mensuration 
problems  concerning  wall-paper  and  carpet  are  rather  futile, 
especially  when  whole  chapters  in  arithmetic  books  are 
devoted  to  them?  Children  are  much  better  employed  in 
mensuration  problems  that  really  do  enter  into  the  practical 
business  of  life. 

Border  Areas 

Make  these  a  matter  of  subtraction,  whenever  possible 
as,  for  instance,  in  estimating  the  area  of  a  garden  path 
4'  wide  between  a  rectangular  lawn  and  the  rectangular 
garden  wall  the  garden  being  108'  X  72'. 

Area  =  {(108  X  72)  -  (100  X  64)}  sq.  ft. 
Do  not  allow  boys  to  find  the  area  of  the  path  piecemeal. 

Rectangular  Solids 

For  the  mensuration  of  these,  boys  can,  with  very  little 
help,  establish  the  necessary  simple  formulae  and  interpret 
them  in  some  brief  form  of  words  easily  remembered.  Prob- 
lems on  the  excavation  of  trenches,  the  cubical  content  of 
cisterns,  the  air  space  of  school  dormitories,  and  the  like, 
will  readily  occur  to  the  teacher.  The  cubical  content  of  a 
solid  "  shell  "  (e.g.  of  iron  in  a  cistern,  of  stone  in  a  rectangular 
trough)  should,  whenever  possible,  be  made  a  problem  of 
subtraction.  Example:  Find  the  weight  of  a  stone  trough 
6"  thick,  external  dimensions  10'  X  3'  X  2'  6",  the  weight  of 
stone  being  \\  cwt.  to  1  cubic  foot. 

No.  of  c.  ft.  of  stone  =  (10  X  3  X  2J)  -  (9  X  2  X  2) 

=  75-36 

=  39. 
Weight  =  1J  cwt.  X  39  =  2  tons,  18|  cwt. 

A  boy  should  never  attempt  to  cube  up  the  stone  piecemeal. 

If  a  gasholder  ("  gasometer  ")  at  an  ordinary  gas-works  is 

made  the  subject  of  a  mensuration  problem,  remember  that 


122  CRAFTSMANSHIP  IN  MATHEMATICS 

(1)  a  gasholder  has  no  bottom,  (2)  its  top  is  not  flat.    Not  all 
writers  of  arithmetic  books  seem  to  realize  this. 

Mensuration  beyond  the  very  elementary  stage  is  best 
associated,  primarily,  with  the  geometry  rather  than  with  the 
arithmetic. 


CHAPTER  XV 

The   Beginnings  of  Algebra 

Informal  Beginnings 

Regarded  as  simple  generalized  arithmetic,  algebra  will 
have  been  begun  at  the  age  of  about  9  or  10.  Quite  young 
boys  will  have  measured  up  rectangular  areas  and  will  have 
learnt  to  express  intelligently  the  meaning  of  the  formula 
A  =  /  X  b.  In  their  lessons  on  physical  measurements, 
rather  older  boys  will  probably  have  evaluated  77,  2?7R, 
?rR2;  in  their  arithmetic  lessons  they  will  have  established 

PTR 

the  formula  I  =  ;    in  their  first  lessons  on  Ratio  and 

x          1 

Proportion,  they  will  have  learnt  the  significance  of  -  =   — 

6         14 

and  will  have  obtained  the  first  notions  of  an  equation. 
Formally,  algebra  will  not  have  been  begun;  informally, 
foundations  will  have  been  laid. 

Never  begin  the  teaching  of  the  subject  according  to  the 
sequence  of  the  older  textbooks.  The  difficult  examples  in 
mechanical  work  so  often  given  on  the  first  four  rules,  on 
H.C.F.s  and  L.CJVI.s,  on  fractions,  &c.,  are  not  only  calculated 
to  make  boys  hate  the  subject  but  are  wholly  unprofitable 
either  at  an  early  stage  or  later. 

Suppose  you  are  asking  questions  in  mental  arithmetic 
to  a  class  of  boys  10  or  11  years  of  age,  and  you  suddenly 
spring  upon  them  the  sum,  "  add  together  all  the  numbers 


THE  BEGINNINGS  OF  ALGEBRA  123 

from  1  to  100."— "  We  cannot  do  it,  sir/'—"  Well,  let  us 
try.  Let  us  first  take  an  easier  sum  of  the  same  kind:  add 
together  all  the  numbers  from  1  to  12.  We  will  do  it  in  this 
way: 

"  Add  together  the  first  number  and  the  last,  1  and 

12?    13. 
"  Add  the  2nd  from  the  beginning  and  the  2nd  from 

the  end,  2  and  11?    13. 
"  Add  the  3rd  from  the  beginning  and  the  3rd  from 

the  end,  3  and  10?    13. 
"  4  and  9?    13;    5  and  8?    13;    6  and  7?   13. 

Now  we  have  included  them  all.  How  many  13's?  6.  What 
aresix!3's?  78.  This  78  must  be  the  answer  to  the  question. "{ 
Smiles  of  agreement. 

"  Now  let  us  make  up  a  little  formula  that  we  can  use 
for  similar  sums:  How  did  we  obtain  the  first  13?  We  added 
together  the  first  number  and  the  last. 

"  What  is  the  first  letter  of  the  word  first?  f. 
"  What  is  the  first  letter  of  the  word  last?   I. 
"  How  can  we  show  the  sum  of/  and  /?  /  +  /. 
"  How  far  along  the  line  from  1  and  12  was  our  mul- 
tiplier, 6?    Half-way. 
"  What  is  the  first  letter  of  the  word  half?  h. 

"  Now  I  will  show  you  how  to  write  down/  +  /  multiplied 
by  h"  Then  follows  a  brief  explanation,  and  A(/+  0- 

"  Now  let  us  work  the  harder  sum,  1  to  100. 

"/=  ?»    1;  "/=  ?»    100;    "A  =  ?"   50. 

"  .'.  h(f  +  /)  =  50(1  +  100)  =  (50  X  101)  =  5050. 

Now  add  together  all  the  numbers  from  1  to  1000." — And 
so  on.  "  I  have  been  giving  you  an  algebra  lesson  which  I 
sometimes  give  to  boys  2  or  3  years  older.  An  interesting 
subject,  isn't  it?"  Yes.  "  And  useful?"  Yes. 

Or  we  might  begin  straight  away  with  problems  producing 
equations.  First  notions  of  an  equation  will  already  have 


124  CRAFTSMANSHIP  IN  MATHEMATICS 

been  given  in  arithmetic.  By  means  of  a  few  easy  exercises, 
revise  the  principle  that  the  two  sides  of  an  equation  may 
be  added  to,  diminished,  multiplied,  or  divided,  by  any 
number  we  please,  provided  that  the  two  sides  are  treated 
exactly  alike.  The  rule  of  cross  multiplication  should  also 
be  revised.  But  naturally  at  thi$  stage  no  equation  should 
be  given  with  a  binomial  in  a  denominator. 

Here  are  two  examples,  in  a  teacher's  own  phraseology 
(summarized,  except  that,  to  save  space,  his  many  admirably 
framed  questions  are  omitted),  once  taken  with  a  class  of 
beginners  of  11. 

The  sum  of  £50  is  to  be  divided  among  2  men,  3  women, 
and  4  boys,  so  that  each  man  shall  have  twice  as  much  as  each 
woman  and  each  woman  3  times  as  much  as  each  boy.  Required 
the  share  of  each. 

"  In  sums  of  this  kind  it  is  always  well  to  consider  first 
the  person  who  is  to  have  the  least,  in  this  case  a  boy. 

Let  x  be  the  number  of  £'s  in  each  boy's  share. 
Then  3x  is  the  number  of  £'s  in  each  woman's  share. 
And  Qx  is  the  number  of  £'s  in  each  man's  share. 
Hence  we  have,  in  £'s, 

The  share  of  the  4  boys  =  4#. 
The  share  of  the  3  women  =  9x. 
The  share  of  the  2  men  =  I2x. 

But  the  sum  of  all  these  shares  amounts  to  £50. 

.'.  4*  -f-  9*  +  12*  =  £50, 
25*  -  £50, 


/  want  to  divide  some  nuts  among  a  certain  number  of 
boys.  If  I  give  4  nuts  to  each  boy,  I  shall  have  2  nuts  to  spare; 
if  I  give  3  to  each  boy,  I  shall  have  8  to  spare.  How  many  boys 
are  there? 

"  Let  x  be  the  number  of  boys. 

"  There  are  two  parts  to  this  problem,  both  beginning 
with  the  word  if\  each  part  enables  us  to  write  down  the 


THE  BEGINNINGS  OF  ALGEBRA  125 

total  number  of  nuts>  though  not  exactly  as  in  arithmetic, 
because  we  have  to  use  x. 

"  1.  If  I  give  4  nuts  to  each  of  x  boys,  I  give  away  4# 
nuts.     But  the  total  number  of  nuts  is  2  more  than  that. 

.'.  the  total  number  of  nuts  =  4#  -f-  2. 

"  2.  If  I  give  3  nuts  to  each  of  x  boys,  I  give  away  3x 
nuts.    But  the  total  number  of  nuts  is  8  more  than  that. 

/.   the  total  number  of  tints  --  3#  -f  8. 

.'.   4x  +  2  =  3jc  +  8, 

.'.     *  +  2  =  8, 


This  class  of  boys  subsequently  spent  the  next  three  or  four 
lessons  in  working  through  a  chapter  of  problems  (some  of  them 
pretty  difficult)  producing  equations.  They  gained  confidence 
quickly,  and  from  the  first  looked  upon  their  new  subject  as 
interesting  and  useful. 

Formal  Beginnings.     Signs  as  "Direction  Posts" 

Of  course  the  time  comes  when  algebra  must  be  treated 
formally.  There  are  certain  fundamental  difficulties  that  have 
to  be  faced,  and,  of  these,  algebraic  subtraction  is  to  beginners 
a  difficulty  of  a  serious  kind. 

This  does  not  mean  that  boys  need  be  taken  through  the 
elaborate  subtraction  sums  of  the  older  textbooks.  It  means 
that  they  have  to  be  taught  the  inner  meaning  of,  say,  "  take 
—  3#  from  —  2#  ".  To  the  boy,  what  does  that  meant  At 
first  it  can  mean  nothing  but  juggling  with  arithmetical 
values,  juggling  of  which  he  is  naturally  suspicious. 

Consider,  first,  what  the  boy  has  already  done  (or  ought 
to  have  done)  in  his  arithmetic.  He  is  familiar  with  this  kind 
of  sum.  17  _  6  __  5  +  3  _  9  +  14 

=  17  +  3  +  14-6-5-9 
=  (17  +  3  +  14)  -  (6  +  5  +  9) 
=  34-20 
=  14. 


126  CRAFTSMANSHIP  IN  MATHEMATICS 

He  has  been  taught  to  collect  up  his  numbers  in  this  way, 
and  to  realize  that  the  plan  of  adding  together  all  the  minus 
numbers  and  taking  them  away  "in  a  lump"  is  a  much  better 
plan  than  taking  them  away  separately.  Thus  he  sees  that 

34-6-5-9 

must  be  the  same  as 

34  -  (6  -f  5  +  9), 

and  he  therefore  gets  a  first  notion  of  the  effect  of  a  minus  sign 
before  a  pair  of  brackets.  Still,  the  work  so  far  is  wholly 
numerical,  and  nothing  more.  If  the  sum  had  been 

20-  34 

he  would  probably  have  been  taught  to  prefix  a  +  sign  to 
the  20: 

+  20  -  34, 

to  take  the  difference  between  the  20  and  the  34,  and  to  prefix 
the  sign  of  the  larger  number  (34);  thus,  —14.  Naturally 
the  boy  would  call  it  a  "  subtraction  "  sum,  and  would  say 
that  the  —14  is  'Mess  than  nothing  "I  The  teacher  might, 
however,  allow  the  use  of  this  illogical  expression  provisionally, 
comparing  it  with  debts  as  against  assets. 

But  the  boy  must  soon  come  to  grips  with  the  funda- 
mental algebraic  notion  of  direction,  as  well  as  of  numerical 
value. 

In  arithmetic  always,  in  algebra  commonly,  the  +  sign 
before  an  initial  term  is  omitted.  But  in  the  early  stages  of 
algebra  it  is  advisable  that  it  be  consistently  written. 

In  algebra  it  is  necessary  to  have  some  means  of  distin- 
guishing direction  to  the  right  from  the  opposite  direction 
to  the  lefty  and  direction  upwards  from  the  opposite  direction 
downwards.  The  opposed  signs  +  and  —  are  used  for  this 
purpose.  These  signs  are  the  algebraic  signposts  or  direction 
posts;  the  two  signs  direct  the  numbers  to  which  they  are 
attached.  It  has  been  agreed  that  direction  upwards  and  to 
the  right  shall  be  called  a  +  (positive)  direction,  and  direction 


THE  BEGINNINGS  OF  ALGEBRA 


127 


downwards  and  to  the  left  a  —  (negative)  direction.  The 
converse  would  have  done  equally  well,  but  the  decision  has 
been  universally  accepted.  It  is  just  a  convention.  If  the 
boy  asks  why?  tell  him  there  is  no  answer. 

Consider  the  centigrade  thermometer,  with  the  freezing- 
point  marked  0°.  If  the  temperature  is  —5°  and  rises  20°, 
every  boy  knows  that  it  rises  to  +  15;  i.e.  —-5  +  20  =  +15; 
also  that  if  the  temperature  is,  say,  +10°,  and  falls  25°, 
it  falls  to  —15°,  i.e.  +10  —  25  —  —15.  The  results  may  be 
obtained  by  actual  counting,  upwards  or  downwards,  on  the 
scale.  Upward  counting  means  adding  +  numbers;  down- 
ward counting  means  adding  —  numbers.  The  thermometer 
provides  an  excellent  means  of  giving  a  first  lesson  on  algebraic 
direction. 

Addition  and  Subtraction 

Now  consider  a  more  general  case.  We  will  choose  a 
horizontal  scale,  with  +  numbers  and  —  numbers  to  the 
right  and  left,  respectively,  of  a  zero. 

Adding  +  quantities  means  counting  to  the  right. 
Adding  -—  quantities  means  counting  to  the  left. 


__ — . — ___ — |— j— p-j 

-10    -9    -8    -7    -6    -5    -4    -3    -2    -1 


I        I        1        I        I        I        I        I        I       I 
+1   +2   +3   4-4   +5   +6  +7   +8  +9   +10 


Four  addition  sums: 


Addition  Sums. 

Starting  Point 
on  Scale. 

Count  or 
Add  on  Scale. 

Direction, 
Ror  L. 

New  Point  on 
Scale  and  /.  Ans. 

i 

(+5)  +  (+3) 

+3 

+5 

R 

+  8 

ii 

(  +  5)   +   (-3) 

-3 

+  5 

R 

+2 

iii 

(-5)  +  (+3) 

+3 

-5 

L 

2 

iv 

(-5)  +  (-3) 

-3 

-5 

L 

-8 

Four  subtraction   sums. — Where   we   have   to   work   a 
subtraction  sum,  say  12  —  7,  we  may  work  it  by  asking  what 


128 


CRAFTSMANSHIP  IN  MATHEMATICS 


we  must  add  to  7  to  make  12.    Thus  in  the  four  subtraction 
sums  below  we  may  say, 

(i)  What  must  we  add  to  (+3)  to  make  (+5)? 
(ii)  ,,  „  ,,  (-3)  „  (+5)? 
(Hi)  ,,  „  „  (+3)  „  (-5)? 
(iv)  „  „  „  (-3)  „  (-5)? 


Subtraction  Sums. 

Starting  Point 
on  Scale. 

Scale  Point 
to  count  to. 

Direction, 
R  or  L. 

Number  of  Points 
counted 

=  Answer. 

i 

(  +  5)   -   (  i  3) 

+  3 

-1-5 

R 

+  2 

ii 

(  +  5)   -   (-3) 

-3 

-1-5 

R 

+  8 

iii 

(-5)   -   (  +  3) 

+  3 

-5 

L 

-8 

iv 

(-5)  -   (-3) 

-3 

-5 

L 

-2 

i 

Let  the  boys  now  examine  the  two  groups  of  answers  and 
note  from  them  that: 


(  +  5)  +  (  +  3)  - 

(  +  5)  +  (-3)  = 

(-5)  +  (  +  3)  = 

(-6)  +  (-3)  = 


(  +  6)  -(-3) 
(  +  5) -(4-3) 
(-5) -(-3) 
(-5) -(4-3) 


They  thus  learn  that  in  every  case  we  can  turn  a  subtraction 
sum  into  an  addition  sum  merely  by  changing  the  sign  of  the 
subtrahend. 

They  ought  now  to  understand  that  in  arithmetical  addition 
the  total  is  increased  by  each  term  added;  that  in  algebraic 
addition  the  numbers  indicate  movements  or  distances  back- 
wards or  forwards  along  a  line  from  a  zero  point  or  "  origin  ". 

They  ought  also  to  see  that  in  algebraic  addition  we  may 
drop  the  sign  which  separates  the  components,  and  deal 
with  the  components  in  accordance  with  their  own  signs,  e.g. 

(4-5)  +  (4-3)  -4-54-3 
(4-5)  4-  (-3)  =  +5-3 
(-5)  +  (  +  3)-  -5  +  3 
(_5)  +  (-3)=  -5-3 


For  algebraic  subtraction,  let  them  substitute  algebraic 


THE  BEGINNINGS  OF  ALGEBRA  129 

addition,  at  the  same  time  always  reversing  the  sign  of  the 
second  term  (subtrahend).  Since  the  sum  is  now  an  addition 
sum,  we  may  drop  the  connecting  +  sign  as  before: 

(  +  5)  -  (  +  3)  =  (  +  5)  +  (~3)  =  +5  -  3 
(  +  5)  -  (-3)  =  (  +  5)  4-  (  +  3)  -4-5-1-3 
(-5)  -  (4-3)  -  (-5)  4-  (-3)  =  -5  -  3 
(-5)  -  (-3)  -  (-5)  4-  (4-3)  --64-3 

Examples:    Add      11  x  and  — 19*: 

4-17*  -  19*  =  -2*. 
From   17*  take  — 19*: 

4-17*  -  (-19*)  -  17*  4-  19*  =  4-36*. 

For  the  slow  boys,  indeed  for  all  boys,  the  whole  process 
crystallizes  into  three  simple  little  rules: 

1.  Addition  sums. 

(i)  Like  signs:   add,  and  prefix  the  same  sign, 
(ii)  Unlike  signs:    find  the  difference  between  the  two 
numbers  and  prefix  the  sign  of  the  larger. 

2.  Subtraction  sums.     Reverse  the  sign  of  the  second  term 

(subtrahend)  and  treat  the  sum  as  an  addition  sum. 

Teachers  are  not  always  quite  happy  about  this  question  of 
directed  numbers,  and  often  ask  if  it  is  not  unwise  even  to 
make  the  attempt  to  deal  with  it,  and  if  a  statement  of  just  the 
rules  ought  not  to  suffice.  Of  the  answer  I  have  no  doubt. 
Boys  who  do  not  grasp  the  significance  of  directed  numbers 
can  never  get  to  the  bottom  of  their  algebra;  their  work 
all  through  will  inevitably  be  mechanical.  Admittedly, 
however,  the  non-mathematical  boy  fails  to  understand,  and 
for  him  the  rules,  as  rules,  must  suffice.  A  Sets  can  and  must 
master  the  difficulty,  and  I  think  B  Sets  too.  But  with  C  Sets, 
and  especially  with  D  Sets,  be  content  to  state  the  rules 
and  to  give  the  boys  plenty  of  practice  in  them.  Such  boys 
will  never  make  mathematicians,  and  nobody  expects  that  they 
will.  It  is  best  to  admit  that  the  application  of  signs  to  com- 
ponent and  resultant  scale  distances  is  a  little  too  subtle 
for  the  non-mathematical  boy. 

(E291)  10 


130 


CRAFTSMANSHIP  IN  MATHEMATICS 


M 


+200 


-5    -4    -3     -2 


+2    +3 


+5 


Multiplication 

Here  again  the  rule  of  signs  can  be  understood  only  by 
a  clear  grasp  of  the  effect  of  direction.  The  usual  train 
illustration  is  as  good  as  any.* 

Graph  the  route  of  a  train  travelling  northwards  through  O 
(say  Oxford)  at  the  rate  of  40  miles  an  hour,  and  thus  show 
the  position  of  the  train  at  all  points  on  its  journey. 

Let  horizontal  lengths  to  the  right  of  MOM'  measure 
times  after  train  reaches  O,  and  let  the  times  be  indicated 

by  -f-  numbers;  and 
let  those  to  the  left 
of  MOM'  measure 
times  before  train 
reaches  O,  and  let 
these  times  be  indi- 
_H  cated  by  —  numbers. 
Let  lengths  above 
H'OH  measure  dis- 
tances north  of  O, 
and  let  these  be  indi- 
cated by  +  numbers; 
and  let  those  below 
H'OH  measure  dis- 
tances south  of  O, 
and  let  these  be  indicated  by  —  numbers. 

We  will  mark  the  positions  of  the  train  4  hours  before 
reaching  O  and  4  hours  after  passing  O.  (The  scales  used 
are  5  mm.  to  50  miles  and  5  mm.  to  1  hour.)  At  40  miles 
an  hour,  the  train  must,  at  these  times,  be  160  miles  short  of 
O  and  160  beyond  O,  respectively.  Plot  points  P2  and  Pl 
to  show  this.  P2  must  be  directly  below  —4  on  H'H,  and  to 
the  left  of  —160  on  M'M;  Pl  must  be  directly  above  +4 
on  H'H  and  to  the  right  of  +160  on  MM'.  The  line  P2Pa 
evidently  passes  through  O,  and  represents  the  train  route. 


P2(-I60) 


-60 


-ISO 


-200 

M' 

Fig. 14 


•See  Nunn,  Teaching  of  Algebra,  Chap.  XVIII. 


THE  BEGINNINGS  OF  ALGEBRA 


131 


Now  how  can  we  determine  the  two  positions  by  cal- 
culation? 

We  may  utilize  the  formula  d  =  vt  ("  distance  =  speed 
X  time  "),  and  by  making  the  three  symbols  stand  for 
directed  numbers,  the  formula  will  give  us  information  about 
the  direction  as  well  as  the  magnitude.  Hence  we  must  use 
the  term  velocity.  Let  velocity  northwards  (40  miles  an  hour) 
be  considered  -f. 

1.  Position  of  train  at  P2: 

d  =  vt  =  (  +  40)  x  (-4) 

=  -160  (as  graphed)  =  160  miles  S. 

2.  Position  of  train  at  Px: 

d  =  vt  =  (  +  40)  x  (  +  4) 

=  +160  (as  graphed)  =  160  miles  N. 

Now  consider  the  train  travelling  southwards.  Let  velocity 
southwards  (40  miles  an  hour)  be  considered  negative  (— -). 


3.  Position  of  train  at  P3: 

d  =  vt  =  (-40)  X  (-4) 

=  +160  (as  graphed)  =160  miles  N. 

4.  Position  of  train  at  P4: 

d  =  vt  =  (-  40)  x  (  +  4) 

=  -160  (as  graphed)  =  160  miles  S. 


132  CRAFTSMANSHIP  IN  MATHEMATICS 

Comparing  the  4  results  we  have: 

(  +  40)  x  (  +  4)  =  +160 

;  +  40)  x  (-4)  =  -160 

(-40)  X  (-4)  ==  +160 

(-40)  X  (  +  4)  =  -160 

This  is  enough,  at  this  stage,  to  justify  the  sign  rule  for  mul- 
tiplication. A  more  rigorous  generalization  may,  if  necessary, 
come  later. 

(The  boys  should  be  made  to  see  that  the  sloping  lines  in 
the  above  graphs  do  not  graphically  show  the  actual  railway 
track,  which  is  supposed  to  run  due  north-south.) 

It  may  be  urged  that  the  whole  thing  seems  to  be  a 
little  artificial.  So  it  is.  But  the  rule  of  signs  is  a  universally 
accepted  convention.  The  convention  is  perfectly  self- 
consistent,  and  is  easily  justified,  but  by  its  nature  it  admits 
of  no  "  proof  ". 

Book  to  consult:   The  Teaching  of  Algebra,  Nunn. 


CHAPTER  XVI 

Algebra:    Early  Links  with  Arithmetic 
and  Geometry 

Algebra  and  Arithmetic  in  Parallel 

Get  the  boys  to  see  that  an  algebraic  fraction  is  only  a 
shorthand  description  of  actual  arithmetical  fractions,  and 
that  there  is  really  no  difference  in  the  treatment.  The  working 
processes  are  practically  identical. 

The  arithmetical  fraction  -^  may  be  written  —?, 
which  shows  clearly  that  the  denominator  is  greater  by  3 
than  the  numerator.  So  does  the  fraction  — ^,  and  that 
is  all  it  means.  Thus  in  the  fraction  ~j,  a  +  7  represents 


ALGEBRA:    EARLY  LINKS 


133 


a  single  number;  as  in  arithmetic,  it  must  be  moved  as  a 
whole  from  one  place  in  the  expression  to  another.  In  algebra, 
beginners  sometimes  forget  this,  and  treat  the  parts  of  a 
binomial  denominator  separately.  So  with  2  or  more  bi- 
nomial denominators:  for  instance  in  - — r>4"jirfi>  x  —  ^ 
and  x  +  11  express  single  numbers. 

Show  a  few  corresponding  arithmetical  algebraic  processes 
side  by  side.    It  helps  the  slower  boys  much. 

1.  Let  a  =  4,  6  -•=  7. 


28 

L1 

28" 


28 


- 

a       b 

b  .,.  - 

ttb       ab 


2.  Let  a  =  3,  b  =  4,  c  =  5 

1  +  -4  +  -5 
20        15        12 

=  3»        42        6* 

~~  60       60       60 
32  4.  42  +  52 


3.  Let  a  =  7,  b  —  4. 
_1  J^ i< 

7~- 4       7~+~i    '  49~~ 
=  1        1   _  14 
~  3       11       33 

-  I1  4-    3  _  U 
33       33       33 

-  H  +  3  ~  14 

"33" 

=  JO 

""  33 

-  0. 


be 


4-        4-    ^ 
ca       a6 

6^       ^ 
«^c       a^ 


1        ,        1 


2a 


a  —  b       a  -\~  b       a2  —  b2 


4-  __1_    _  ___ 
a2  -  b*      a*  -  b* 

-^a  —  b  —  2a 


0. 


134 


CRAFTSMANSHIP  IN  MATHEMATICS 


There  is  little  or  no  need  to  take  fractions  beyond  quite  simple 
binomial  denominators.  Denominators  of  a  higher  order  are 
seldom  required  in  practice.  Hence  all  H.C.F.s  and  L.C.M.s 
should  be  evaluated  by  factorizing,  exactly  as  in  arithmetic. 
The  principle  of  the  cumbrous  divisional  processes  for  finding 
factors  should  be  familiar  to  boys  in  A  Sets,  who,  however, 
may  be  referred  to  their  textbooks.  Do  not  waste  time  over 
such  things  in  class. 

Geometrical  Illustrations 

Factors,  multiplication,  division,  simple  expansions,  &c., 
should  all,  in  the  early  stages,  be  illustrated  geometrically, 
and  thus  be  given  a  reality.  When  this  reality  is  appreciated, 
but  not  before,  the  illustrations  may  be  given  up.  Second 
power  expressions  should  be  consistently  associated  with  areas. 
We  append  a  few  illustrative  examples. 


2 

20 

0                  20 

X 

4 

\                 20  *  4 

4* 

2 
2                                ^ 

at 

>                     ab 

b* 

Fig.  1 6 

1.  Compare  the  square  of  24  (i.e.  20  +  4)  with  the  square 
of  a  +  b. 

24a  (a  +  6)a 

=  (20  +  4)8  =fl2  +  2a&  +  62. 

=  202-f  2.20.4  +  42. 

2.  Compare  24x27,  i.e.  (20  +  4)  (20  +  7),  with 


ALGEBRA:    EARLY  LINKS 


135 


(20  +  4)  (20  +  7) 

=  20a  +  20.7  +  20.4  +  4.7 
=  202  +  20(7  +  4)  +  4.7. 


(a  +  4)  (a  +  7) 
=  a2  +  7a  +  4a  +  4.7 
=  a2  +  a(7  +  4)  +  4.7. 


-20- 


202 

20 

X 

4 

20*7 

4 

X 

7 

2 

CL 

I 

a 

X 

4 

r 

\ 

a  x  7 

4 
x 
7 

Fig.  17 


3.  Show  graphically  that 

(2a  +  56)  (a  +  36)  =  2a2  +  Uab  +  1562. 

o.  a.  b      b      b      b      b 


a 

b 
b 
b 


Fig.  1 8 

The  result  is  seen  at  a  glance. 


2 
a. 

2 

a 

ab 

ab 

. 

ab 

. 

I 

an 

cxb 

ab 

b2 

b2 

b2 

b2 

b2 

db 

ob 

b? 

b2 

b2 

b2 

b2 

ab 

ab 

b2 

b2 

b2 

b2 

b2 

1 

?<,  +  5^ 

136  CRAFTSMANSHIP  IN  MATHEMATICS 

4.  Show  graphically  that  (a  —  2)  (a  —  3)  =  a2  —  6a  +  6. 


A 

•*—  a  -2  *• 

/ 

D 

t 
a.  S 

i 

F 

J 
a 

G 

E 

/ 

/ 

f 

B 

'*                       Q- 

C 

AE  =  (a  -  2)  (0  -  3), 
AC  -  a2, 
GC  -  3a, 


.  19 

rcc\-l 
==   ^<z, 

GC  +  FC  -  EC  =  6fl  -  6, 
AE  -  AC  -  (GF  -f  FC  -  EC) 


i.e.  (a  -  2)  (a  -  3)  =  a2  -  5a  -f  6. 
5.  Show  graphically  that  (a  +  2)  (a  —  3)  —  a2  —  a  —  6. 


4 

E   /      / 

3 

p 

R 

(a  +  2)    (a  -  3> 

t 

a-3 

P 

K             C 

F 

q 

/        /       /   S 

Fig.  20 

(a  +  2)  (a  -  3)  =  AC  I  =  AK  +  PF 

=  AK  f  EC  I  =  AF  -  DQ  -  RS; 

i.e.  (a  +  2)  (a  —  3)  ==  a2  —  a  —  6. 


ALGEBRA:   EARLY  LINKS 

6.  Show  graphically  that 

(a  +  b  +  cf  =  a*  +  b*  +  c2  +  2ab  +  2ac  +  2bc. 


137 


-  —  a.  *|«_b-*|4C*. 

r 

a 

2 
CL 

ab 

ac 

5 

ab 

tf 

be 

[ 

I 

f 

ac 

be 

c2 

Fig.  21 

The  result  is  seen  at  a  glance. 


CHAPTER  XVII 

Graphs 

From  the  Column  to  the  Locus 

Begin  with  column  graphs,  that  is  with  mere  verticals 
with  the  tops  unconnected;  say,  the  amount  of  gas  consumed 
each  week  for  a  quarter,  or  the  height  of  a  barometer  each 
morning  for  a  week.  Now  join  the  tops  of  the  columns, 
first  by  straight  lines,  then  by  a  curved  line.  Do  the  straight 
lines  teach  anything?  Which  is  likely  to  be  the  more  correct, 
the  straight  lines  or  the  curved  line?  Can  intermediate  columns 
be  inserted,  and,  if  so,  what  would  they  signify?  Are  there 
any  cases  where  intermediate  columns  would  be  absurd? 

Now  discuss  the  locus  graph,  as  distinct  from  the  column 


138 


CRAFTSMANSHIP  IN  MATHEMATICS 


graph.  The  barograph  is  a  useful  example  especially  for  the 
contrast  of  a  gentle  slope  and  a  steep  slope,  and  hence  as  an 
introduction  to  a  gradient  and  what  it  signifies.  What  does 
a  chart  of  closely  packed  isobars  signify?  of  open  isobars? 
Or  graph  the  vertical  section  of  a  piece  of  hilly  country,  by 
taking  heights  from  an  ordnance  survey  map.  Here  the  gentle 
slope  and  the  steep  slope  appeal  at  once,  the  closely  packed 
isobars  and  the  closely  packed  contours  being  akin. 


Dates  (20 
of     1^ 
lnnings[  MAY 


The  significance  of  a  gentle  gradient  and  of  a  steep 
gradient  is  fundamental.  It  is  really  the  key  to  all  subse- 
quent work.  Let  the  boys  graph  their  cricket  scores  for 
the  previous  summer  term,  and  discuss  the  resulting 
gradients.  Familiar  and  personal  data  of  this  kind  often 
provoke  animated  discussion  of  a  useful  character.  In  the 
first  lesson  or  two,  much  of  the  work  can  be  done  on  the 
blackboard,  exact  numerical  values  playing  only  a  minor 
part.  Give  the  beginners  a  general  notion  of  the  graph  and 
its  significance.  A  few  instances  may  be  culled  from  chemistry 
and  physics,  say  solution  curves  (common  salt,  with  its 


GRAPHS 


139 


very  slowly  rising  straight  line;  nitre,  with  its  steep  curve); 
the  experimental  results,  with  their  subsequent  pictorial 
illustrations,  are  always  impressive.  Other  useful  graphs 
from  practical  work  are  a  straight-line  graph  from  a  loaded 
spiral  spring,  or  from  a  F.-C.  scale  comparison;  an  inverse 
proportion  graph,  say  a  time-speed  curve  during  a  journey. 
Thus  prepare  the  way  for  formal  work. 


15 


penc 


0 


5 

Ib 

Fig.  23 


A  parcel  post  graph  is  easy  of  interpretation  and,  by 
its  gradient  of  equal  steps,  leads  on  naturally  from  a  column 
graph  to  a  direct  proportion  graph.  It  may  be  called  a 
"  stepped  "  graph.  There  is  a  minimum  charge  of  6d.  for 
any  weight  of  parcel  up  to  2  Ib.;  the  charge  is  9d.  for  any 
weight  over  2  Ib.  and  up  to  5  Ib.;  a  shilling  for  any  weight 
over  5  Ib.  and  up  to  8  Ib.;  and  so  on.  A  straight  line  can 
be  drawn  through  the  corners  of  the  figure,  but  this  straight 
line  does  not  pass  through  the  origin. 

The  mere  plotting  of  a  graph  nowadays  gives  little  trouble. 
Most  modern  books  give  instructions  both  simple  and 
satisfactory.  But  a  clear  understanding  of  what  has  been 
done  and  a  satisfactory  interpretation  of  the  completed 
graph  often  leave  much  to  be  desired.  It  is  the  interpretation 
that  is  the  all-important  thing.  A  graph  is  essentially  a  kind 
of  picture,  a  picture  to  be  understood.  The  pictorial  element 
admits  of  a  general  interpretation  simple  enough  for  be- 


140 


CRAFTSMANSHIP  IN  MATHEMATICS 


ginners  to  understand,  but  as  time  goes  on  this  interpretation 
must  be  made  more  and  more  exacting. 


The  study  of  y  =  tnx  -f  c.    Direct  Proportion 

Experience  convinces  me  that  the  study  of  the  form 
y  =  mx  should  precede  that  of  the  form  xy  —  c.  But  pro- 
portionality of  one  kind  or  another  underlies  the  whole  thing, 
and  the  straight  line  and  rectangular  hyperbola  should  occupy 
a  first  place. 

Do  not  attempt  to  define  for  beginners  the  term  function. 
The  term  should,  however,  be  used  from  the  first.  "  Here 
is  an  expression  involving  x,  that  is,  a  function  of  x."  In 
time,  drop  the  words  "  expression  involving  "  and  simply 
say  "  function  of  ".  Let  the  word  be  used  constantly;  it 
will  gradually  sink  in  and  become  part  of  the  boys'  own 
mathematical  vocabulary. 

Begin  with  a  straight-line  graph  passing  through  the  origin. 

(i)  y  ••=  oc.  What  does  this  mean?  That^y  is  always  equal 
to  x,  i.e.  that  the  ordinate  is  always  equal  to  the  abscissa, 


+3 


+2 


•+I 


•3    -2 


O    +1      +2 
-I 

-2 
-3 


no  matter  what  point  on  the  line  is  taken,  whether  in  the  first 
or  third  quadrant.  Thus  in  the  figure  we  have  the  point 
(3,  3)  in  the  first  quadrant  and  the  point  (—2,  —  2)  in  the 
third. 


GRAPHS 


141 


(ii)y  ==  — -oc.  This  is  practically  the  same  as  before.  The 
length  of  y  is  equal  to  the  length  of  #,  i.e.  the  length  of  the 
ordinate  is  equal  to  the  length  of  the  abscissa,  but  now  the 
signs  are  different,  whether  a  point  is  taken  in  the  second 


-a; 


Fig.  25 

quadrant  (as  —3,  3)  or  in  the  fourth  (as  2,  —2).  The  graph 
runs  from  the  left  downwards,  from  the  second  to  the  fourth 
quadrant. 

(iii)  3y  =  2x.  This  means  that  three  times  the  length  of 
tj|e  ordinate  is  equal  to  twice  the  length  of  the  abscissa.  We 
may  write,  more  simply,  y  =  \x,  and  then  we  see  that  the 


(3,2) 


Fig.  26 


ordinate  is  always  f  of  the  abscissa.    This  is  easily  seen  from 
any  pair  of  values  (save  0,  0)  in  a  table: 

x  =   -.3       _2  0       +3       4-4          4£ 


y  =  _2       -1 


0       4-2       4-2} 


142 


CRAFTSMANSHIP  IN  MATHEMATICS 


No  matter  what  point  in  the  line  is  chosen,  the  ratio  of  (1) 
the  -Lr  to  the  x  axis  to  (2)  the  intercept  on  the  x  axis,  i.e. 

the  ratio  ~,  is  always  |.    This  ratio  is  constant;  the  triangles 

x 

formed  by  drawing  perpendiculars  are  all  similar.  The  slope 
of  the  line  is  always  the  same,  i.e.  the  gradient  of  the  graph 
is  constant. 

(iv)  2y  =  —3x  or  y  —  —  \x.     Here  the  length  of  the 
ordinate  is  always  1^  times  the  length  of  the  abscissa,  but 

+  3 
+  2 

+  I 
-fl     +2   +3 


-2    -/ 


the  two  are  of  opposite  signs,  as  may  be  seen  from  any  pair 
of  values  (save  0,  0). 

x  =    -2     -1         0         2         3  4 

y  =        3          1J       0     -3     -4J     -6 

The  graph  runs  from  the  left  downwards,  from  the  second 
to  the  fourth  quadrant. 

Before  proceeding  further,  give  the  class  plenty  of  mental 
work  from  the  squared  blackboard,  using  a  metre  scale  or 
a  rod  to  represent  the  graph,  holding  it  in  various  positions 
but  always  passing  through  some  selected  named  point  and 
through  the  origin,  and  asking  the  class  to  name  the  equations. 


GRAPHS  143 

I  have  known  a  class  of  thirty  boys  give  almost  instant 
response,  one  after  the  other,  when  tested  in  this  way. 

See  that  the  boys  become  thoroughly  familiar  with  the 
difference  between  y  =  mx  (same  signs,  slope  from  left 
upwards)  and  y  =  — mx  (opposite  signs,  slope  from  left  down- 
wards). Also  see  that  they  are  not  caught  by  the  alternative 
forms  to  these,  viz.  y  —  mx  =  0,  y  -f-  mx  =  0. 

The  next  step  is  to  see  that  the  boys  understand  the  sig- 
nificance of  the  wi  in  the  equation  y  =  mx.  They  already 
know  that  when  the  coefficient  of  y  is  unity,  the  coefficient 

of  x  is   a   ratio   representing  •?-,  i.e.  the  "  steepness  ",  the 

x 

"  slope  ",  or  the  "  gradient  "  of  the  graph,  and  they  are  thus 
prepared  for  the  general  method  of  writing  this  ratio,  viz. 
by  the  letter  m.  Do  not  begin  with  the  general  form  m9  and 
say  that  it  represents  the  slope  of  the  line,  and  then  illustrate 
it  with  numerical  examples.  Begin  with  the  numerical 
examples,  in  order  that  the  boys  may  really  understand  the 
principle;  then  introduce  the  m  as  a  sort  of  shorthand  registra- 
tion of  facts  which  they  already  know. 

The  next  step  is  to  move  the  graph  about  parallel  to 
itself,  and  to  study  the  effect  upon  the  written  function; 
and  so  lead  the  boys  to  see  that  a  graph  which  does  not  pass 
through  the  origin  necessarily  cuts  off  pieces  (intercepts) 
from  both  axes  (we  neglect  the  case  of  a  graph  parallel  to 
an  axis).  We  may  begin  by  graphing  a  few  particular  cases 
of  the  function  y  =  mx  ±  £>  say  y  —  f  #  ±  c: 

y  =  %x  +  2 

y  =  1*  +  1 

y  =  t* 

y  =  f*  -  1 

y  =  f*  -  2. 

Show  the  pupils  how  to  tabulate  two  or  three  pairs  of  values 
of  each  case,  and  how  then  to  draw  the  graphs.  They  may 
then  compare  their  results. 


144 


CRAFTSMANSHIP  IN  MATHEMATICS 


They  will  readily  discover  that  the  +2,  +1,  0,  —  1,  —2, 
represent  merely  the  number  of  units  the  graph  has  been 
raised  or  lowered  (the  third  case,;y  —  §#,  being  an  old  friend). 
The  function  proper,  y  =  f#,  is  the  same  in  all  cases\  the 
slope  is  constant;  the  five  lines  are  parallel.  A  perpendicular 
(ordinate)  dropped  from  any  point  on  the  graph  to  the  x  axis 


\oc+2 


Fig.  28 

shows  a  right-angled  triangle  similar  to  all  other  similarly 
drawn  triangles.  In  every  case,  the  ratio  of  the  sides  round 
the  right  angle  is  given  by  the  m,  the  coefficient  of  x.  The 
number  (the  c)  added  or  subtracted  represents  merely  the 
bit  of  the  y  axis  intercepted  between  the  graph  and  the  x  axis. 
For  this  reason  we  call  such  bits  of  the  y  axis,  intercepts. 

But  when  we  raise  or  lower  the  graph  above  or  below  the 
origin,  the  graph  really  intercepts  both  axes.  If  the  graph  is 
raised  above  the  origin,  a  portion  of  the  y  axis  above  the 
origin  is  intercepted,  and  a  portion  of  the  x  axis  to  the  left 


GRAPHS 


145 


of  the  origin,  as  well.  If  the  graph  is  lowered  below  the  origin, 
a  portion  of  the  y  axis  below  the  origin  is  intercepted,  and 
a  portion  of  the,  x  axis  to  the  right  of  the  origin,  as  well. 
How  in  each  case  are  the  two  intercepts  related? 

Consider  the  first  of  the  above  five  expressions,  viz. 
y  =  f  •#  +  2.  Instead  of  expressing  y  in  terms  of  #,  we  may 
express  x  in  terms  of  y,  thus: 

v     y  =  %x  -f  2 

/.    3y  =  2*  -f  6 


Here  the  #  intercept  is   —3,  where  we  have  precisely  the 
same  graph  as  before  when  the  y  intercept  was  +2.     The 


-2    -' 


Fig.  29 


+  1 


Fig.  30 


function  is  unaltered.  So  with  the  last  of  the  five  expressions, 
viz.  y  =  f#  —  2.  If  we  express  x  in  terms  of  jy,  we  have 
x  ==  2^  +  ^-  The  a?  intercept  is  +3,  and,  as  before,  the  y 
intercept  is  —  2,  the  graph  being  identically  the  same.  The 
function  is  unaltered,  we  have  merely  expressed  it  differently. 

Generally,  however,  we  express  y  in  terms  of  x,  and  the 
added  or  subtracted  quantity  (the  c)  represents  a  y  intercept. 

The  analogous  results  from  the  function  y  =  — f  x  i  c 
may  now  be  rapidly  dealt  with  in  the  same  way. 

Let  the  pupils  occasionally  check  a  graph  by  means 
of  other  pairs  of  tabulated  values.  For  instance,  from  the 
function  y  -=  f  x  +  2  we  have: 


(K291) 


-3         0     +3     +6     +8 

0     +2     +4     +6     +7} 


11 


146 


CRAFTSMANSHIP  IN  MATHEMATICS 


Consider  the  last  point  (8,  7J),  where  OS  =  8  and  PS  =  7J. 
The  slope  of  the  graph  is  determined  by  the  sides  round 
the  right  angle  of  any  right-angled  triangle  determined 
in  the  manner  aforementioned.  In  the  main  figure  we 
see  two  such  triangles  (shown  also  as  separate  figures 


Fig.  31 

with  the  ordinates,  x  and  j,  in  dark  lines).     The  slope  is 
determined 

PR  ^  y _~-  2  =  7^  —  2  =  2 

RQ  x  8  3' 

-  PS  v  7*          2 

or  by 


either  by 


TS        3  +  x       3  +  8 


-  2 


y      _ 


Hence    we    may    write    either 
The  two  are  identical.  *  '6  '6  +  x       '6 

Beginners  are  apt  to  confuse  the  value  of  m  with  the 
co-ordinates  of  some  arbitrarily  chosen  point;  e.g.  to  take 
the  value  (8,  7|)  of  the  above  point  P,  to  convert  it  into  the 

74 

fraction  -?,  and  to  call  it  m.   It  is  a  thing  that  wants  watching, 
8 

The  boys  ought  now  to  realize  that,  in  y  =  mx  +  c, 
the  c  is  of  little  consequence  compared  with  the  all-important 
m\  and  that  it  may  sometimes  be  convenient  to  ignore  the 
c  and  to  plot  the  graph  in  its  fundamental  form  y  =  mx. 
Since  it  then  passes  through  the  origin,  the  function  is  more 
easily  recognizable. 

The  linear  function  should  thus  provide  the  boy  with  a 


GRAPHS  147 

preliminary  training  to  enable  him  to  see  clearly  how  the 
relation  between  variables  may  be  represented  not  only  in 
equation  form  but  pictorially.  He  should  be  able  to  discover 
the  relation  between  the  variables,  that  is,  to  discover  the 
equation  or  law  connecting  them,  to  discover  what  function 
y  is  of  x,  to  discover  m. 

The  beginner  is  often  perplexed  when  told  that  Ax  +  By 
+  C  =  0  is  the  general  form  of  a  linear  equation.  Why 
those  capital  letters,  he  wonders.  But  if  he  first  sees  that 
his  now  familiar  friend  y  =  —  f x  +  -47-  may  be  written 
3x  +  4y  =  18,  he  will  understand  that  the  new  form  provides 
a  neater  way  of  writing  down  the  function,  though  the  all- 
important  m  no  longer  reveals  itself  so  readily.  "  When 
we  write  this  new  and  neater  form  Ax  -f~  By  +  C  =  0,  the 
only  reason  for  using  capital  letters  is  that  it  enables  us  to 
identify  it  readily.  Other  forms  and  their  specific  uses  you 
will  learn  all  in  good  time.  Why  should  we  not  have  different 
ways  of  writing  down  the  same  function?  May  we  not  weigh 
up  in  the  laboratory  a  piece  of  brass  in  ounces  or  in  grams? 
Convenience  dictates  a  choice  of  method." 

It  is  a  good  general  plan  to  lead  up  to  a  general  form 
through  a  few  particular  examples.  To  spring  suddenly 
upon  a  class  such  a  general  form  Ax  +  By  +  C  =  0,  before 
they  have  been  suitably  prepared,  is  not  the  sort  of  thing 
that  an  experienced  teacher  ever  does. 

Independent  and  dependent  variables  are  terms  to  be 
introduced  gradually.  Make  quite  clear  that  the  x  axis  is 
always  used  for  the  quantity  which  is  under  our  control  and 
is  quite  "  independent  "  of  the  other  quantity,  and  that  for 
this  reason  it  is  given  the  name  independent  variable;  and 
that  the  y  axis  is  used  for  values  calculated  from  the  formula, 
or  for  values  observed  in  experiment,  i.e.  values  which 
" depend "  on  the  selected  and  controlled  x  values,  and  it  is 
therefore  called  the  dependent  variable.  Each  time  we  change 
the  value  of  our  selected  x  quantity,  calculation  or  observa- 
tion gives  us  a  related  y  quantity;  and  the  graph  we  draw  is 
a  picture  .to  show  not  only  how  these  pairs  of  quantities  are 


148 


CRAFTSMANSHIP  IN  MATHEMATICS 


related  but  to  show  that  this  relation  is  the  same  for  every 
pair. 

Another  way  of  expressing  the  connexion  between  the 
two  variables  is  to  say  that  the  dependent  variable  is  a  function 
of  the  independent  variable,  the  latter  being  often  called  the 
argument  of  the  function,  since  we  make  it  the  basis  of  our 
argument.  The  graph  of  an  equation  shows  how  the  function 
varies  as  the  argument  varies  and  is  called  the  graph  of  the 
function;  the  abscissa  is  selected  for  the  argument,  and  the 
ordinate  thus  represents  the  function. 

The  Circle 

There  is  little  to  gain  in  spending  much  time  over  the 
circle,  as  it  will  rarely  be  used  except  to  illustrate  the  solution 
of  such  simultaneous  equations  as  x2  +  y2  =  52,  xy  =  24. 
But  it  does  serve  to  illustrate  simply  how  a  formula  is  affected 
where  the  graph  is  "  pushed  about  ".  We  give  the  same 
circle  in  four  different  positions. 


Fig.  33 


Centre  of  circle  at  origin.    Equation:   x2  +  J>2  =  r2. 

The  centre  is  pushed  1|  units  to  the  right;  its  co-ordinates 
are  (1J,  0).  The  horizontal  of  the  right-angled  triangle  is 
no  longer  x,  but  x  diminished  by  1J. 

Equation:  (x  —  1 J)8  +  jy2  =  r2. 


GRAPHS 


149 


The  centre  is  pushed  2j-  units  up;  its  co-ordinates  are 
(0,  2£).  The  vertical  of  the  right-angled  triangle  is  no  longer 
jy,  but  y  diminished  by  2J. 

Equation:   x2  +  (y  -  21)2  =  r2. 


Fig.  34 


Fig.  35 


The  centre  is  pushed  li  units  to  the  left  and  2J-  units 
up.  The  horizontal  of  the  right-angled  triangle  is  x  +  1|> 
and  the  vertical  is  y  —  2|. 

Equation:  (*  +  H)2  +  (y  —  2£)2  -  r2. 


The  Study  of  ;ry  =  ^.     Inverse  Proportion 

The  direct  proportion  graph  we  found  to  be  a  straight 
line.  The  inverse  proportion  graph  (the  rectangular  hyperbola) 
is  naturally  the  next  for  investigation. 

Let  the  learner  himself  plot  some  simple  case:  "  32  men 
take  1  day  to  mow  the  grass  in  the  fields  of  a  farm.  How 
many  days  would  it  take  16,  8,  4,  and  2  men,  and  1  man  to 
do  it?"  (An  absurd  example,  really,  but  for  our  present 
purpose  the  weather  conditions  and  the  growth  of  the  grass 
may  be  ignored.) 


150 


CRAFTSMANSHIP  IN  MATHEMATICS 


With  half  the  number  of  men,  twice  the  number  of 

days  would  be  required. 
With  one- third  the  number  of  men,  three  times  the 

number  of  days  would  be  required. 

And  so  on.    Hence,  for  graphing,  we  may  write  down  these 
pairs  of  values. 


men  \         32 
days  |          1 


16 
2 


8 
4 


2 
16 


1 
32 


The  graph  is  evidently  a  smooth  curve.  Lead  the  class  to 
discover  that  the  product  of  each  pair  of  values  is  constant, 
that  xy  is  32  in  all  cases. 


Fig.  36 


Now  plot  xy  =  k  for  several  values  of  k,  e.g.  k  —  25, 
49,  64,  100,  225,  400,  and  examine  the  curves  as  a  family. 
How  are  they  related? 

1.  A  line  bisecting  the  right  angle  at  O  divides  all  the 
curves  symmetrically. 

2.  The  point  where  that  line  cuts  the  curve  is  the  point 
nearest  the  origin;  it  is  the  "  head  "  or  vertex  of  the  curve. 


GRAPHS 


151 


3.  At  a  vertex  V,  x  =  y.    Hence,  •  .•  xy  =  ky  x  =  y  = 
.• .  in  *jy  =  25,  the  co-ordinates  of  the  vertex  V  are  (5,  5). 

4.  Each  curve  approaches  constantly  nearer  the  axes,  but 

k 
never  reaches  them.    However  great  the  length  of  x,  y  =  - 

and  y  can  therefore  never  be  zero.  Neither  can  x  ever  be 
zero.  Either  may  be  indefinitely  small  because  the  other 
may  be  indefinitely  large,  but  neither  can  be  absolutely  zero. 
Hence  we  say  that  the  axes  are  the  asymptotes  of  the  curve. 


oc  at  -  25 

d 
ecu  «  49 

DC  of  *  64 
ocu  -;oo 


DC 


-  225 
-400 


/o 


/5      20     25     30 

Fig.  37 


This  term  means  that  the  line  and  the  curve  approach  each 
other  more  and  more  closely  but  never  actually  meet  (asymptote 
=  "  not  falling  together  "). 

5.  The  successive  curves  are  really  similar,  although  at 
first  they  do  not  appear  so.  But  draw  any  two  straight  lines 
through  the  origin  to  cut  the  curves  and  examine  the  in- 
tercepted pieces  of  the  curves  (it  is  best  to  cover  the  parts 
of  the  figure  outside  these  lines),  and  each  outer  bit  of  curve 
will  be  seen  to  be  a  photographic  enlargement  of  the  next 
inner  bit. 

Boyle's  Law  is  the  commonest  example  of  inverse  pro- 
portion in  physics.  But  the  data  (p  and  v)  obtained  from 
school  experiments  are  usually  too  few  to  produce  more 
than  a  small  bit  of  curve,  much  too  small  for  ready  inter- 


152  CRAFTSMANSHIP  IN  MATHEMATICS 

pretation.  But  inasmuch  as  the  law  pv  —  k  seems  to  be 
suggested  by  the  data,  this  may  be  verified  in  two  ways: 
(1)  find  the  product  of  p  and  v  for  each  pair  of  related  values 
and  see  if  the  product  is  constant;  (2)  convert  the  apparently 
inverse  proportion  into  a  case  of  direct  proportion  by  plotting 

not  v  against  p  but  -  against  p.    The  points  thus  obtained 
v 

ought  to  lie  on  a  straight  line,  and  the  line  may  be  tested  by 
means  of  a  ruler,  or  a  piece  of  stretched  cotton.  Does  the 
line  pass  through  the  origin?  Why? 

There  is  probably  little  advantage  in  teaching  boys  to 
"  push  about  "  into  new  positions  the  rectangular  hyperbola, 
though  for  purposes  of  illustration  one  or  two  examples  may 
usefully  be  given.  If  the  graph  xy  ~  120,  or  y  —  ^°,  is 
raised,  say,  3  units,  the  function  becomes  y  =  -^..o  +  3  or 
y  —  3  =  '  jo.  If  it  is  iowered  3  units>  y  _j_  3  ^  .1 .? o.  If 

it  is  raised  3  units  and  then  moved  4  units  to  the  right,  the 
function  reads  y  —  3  =  i~(]  or  y  =  ^« +  3.  But  the 
beginner  is  apt  to  find  this  a  little  confusing.  It  is  best  to 
let  him  keep  the  curve  in  a  symmetrical  position,  and  to 
continue  to  use  the  asymptotes  for  his  co-ordinate  axis. 

Negative  values. — Instruct  the  class  to  graph  xy  =  100 
for  both  positive  and  negative  values.  Then  proceed  in  this 
way. 

11  When  we  plotted  pairs  of  quantities  from  a  linear 
function,  we  passed  from  negative  values  through  the  origin 
to  positive  values  (or  vice  versa),  and  the  graph  was  con- 
tinuous— an  unbroken  straight  line.  Apparently,  then,  the 
rectangular  hyperbola,  though  consisting  of  two  separated 
parts,  ought  to  be  regarded  as  a  single  continuous  curve. 
Is  this  possible? 

"  The  curve  in  the  third  quadrant  is  certainly  an  exact 
reproduction  of  that  in  the  first. 

"  Suppose  the  x  axis  indefinitely  extended  both  ways,  and 
a  point  Z  far  out  to  the  right  to  travel  along  it  towards  O 
the  origin.  At  any  position  it  may  be  regarded  as  the  foot 
of  the  ordinate  of  a  corresponding  point  P  on  the  curve. 


GRAPHS 


153 


As  (for  instance)  Zx  moves  to  Z2,  Px  moves  round  the  curve 
to  P2,  and  as  ZO  diminishes  in  length  (ZjO  to  Z2O),  the 
ordinate  PZ  increases  (P^  to  P2Z2).  But  however  long 
PZ  may  be,  it  gets  still  longer  as  Z  gets  still  nearer  O.  In 
fact,  it  seems  to  become  endlessly  long,  and  yet  we  cannot 
say  that  the  curve  ever  really  meets  the  y  axis,  for  it  is  absurd 


3U 

nc 

Of) 

\ 

XI 

c 

/  ** 

oo 

15 

V 

PZ 

ID 

\ 

^ 

^^ 

Ff 

r 

—  «- 

0    -'< 

—  •    — 

5    -2 

-  

p     -/ 

5     -/ 

3      -: 

>      O 

*-5 

r     i 
•2 

0       1 

5       2 

°z,  2 

J    i 

^ 

k 

-IO 

\ 

•^16 

\ 

~9n 

\ 

f2S 

\ 

---3o 

Fig.  38 

to  speak  of  the  quotient  -$-.  But  if  Z  continues  its  march, 
it  must  eventually  pass  to  the  other  side  of  O.  And  yet  no 
interval  can  be  specified  to  the  left  and  right  of  O  so  short 
that  there  are  no  corresponding  positions  of  P  still  nearer 
to  the  y  axis — on  the  right  at  an  endless  height  and  on  the 
left  at  an  endless  depth.  As  Z  proceeds  along  QJC,  P  simply 
repeats  in  reverse  order  along  the  curve  in  the  third  quadrant 
its  previous  adventure  along  the  curve  in  the  first.  The 
crossing  of  Z  over  the  y  axis  at  O  seems  to  have  taken  P 


154  CRAFTSMANSHIP  IN  MATHEMATICS 

instantaneously  from  an  endless  northern  position  to  an 
endless  southern  position.  We  feel  bound  to  regard  the 
two  curves  as  two  branches  of  the  same  graph,  for  both  are 
given  by  the  function  xy  ~  k  =  100. 

"  If  you  plot  xy  —  —  k,  the  branches  appear  in  the  second 
and  fourth  quadrants." 

The  above  argument  is  always  appreciated  by  A  Sets, 
though  naturally  its  implications  are  too  difficult  for  them 
to  understand  until  later.  With  lower  Sets,  it  is  futile  to 
discuss  the  subject  at  all. 

With  A  Sets,  too,  the  use  of  the  term  "  hyperbolic  func- 
tion "  is  quite  legitimate.  We  called  ax  +  b  a  linear  function 
of  x  because  the  graph  of  y  —  ax  +  b  is  a  straight  line. 
Similarly  we  may  call  any  function  that  may  be  thrown  into 

k 

the  form +  b  a  hyperbolic  function  of  x,  because  the 

x  ~\~  a 

k 

graph  y  =  +  b  is  a  (rectangular)  hyperbola. 

x  -f-  a 

The  Study  of  y  =  x2.    Parabolic  Functions 

The  pupil  should  master  two  or  three  new  principles 
before  he  proceeds  to  the  quadratic  function. 

1.  The  first  is  the  nature  of  a  "root"  of  a  simple  equation. 
A  very  simple  case  will  suffice  to  make  the  notion  clear. 
The  boy  knows  already  that  the  root  of  the  equation  x  —  3  =  0 
is  3.  Now  let  him  graph  the  function  y  =  x  —  3. 

Since  y  —  x  —  3  we  have: 


*  = 

0 

- 

1 

2 

3 

y  =  X  —  3  = 

-3 

-2 

1 

0 

The  line  crosses  the  x  axis  at  +3,  that  is  when  y  =  0,  x  =  3, 
and  we  say  therefore  that  3  is  the  "  root  "  of  the  equation 
#  —  3  =  0.  Of  course  we  should  never  let  a  boy  waste  his 
time  by  actually  solving  an  equation  in  this  manner,  but 
it  serves  to  teach  him  that  when  the  value  of  a  function 


GRAPHS 


155 


equals  0,  then  the  intercept  on  the  x  axis  gives  the  root  of 
the  equation  represented  by  the  function.     (Fig.  39.) 

The  roots  of  related  equations  are  easily  derived.      For 
instance,  solve  the  equation  #—3=1.     (Fig.  40.) 


'3.0  ' 


Fig.  39 


Fig.  40 


Write  y  =  x  -  3  =  1;  i.e.  y  =  x  -  3,  and  y  =  1.  The 
graph  of  y  =  x  —  3  is  the  same  as  before;  the  graph  of  y  =  1 
is  a  line  parallel  to  the  x  axis,  1  unit  above.  The  value  of  x 
in  the  equation  #  —  3  =  1  is  given  by  the  intercept  that 
y  =  x  —  3  makes  with  y  =  1,  i.e.  4.  In  other  words  the  root 
of  the  equation  is  the  x  value  of  the  point  of  intersection  of 
the  two  lines. 

Evidently  we  have  the  clue  for  solving  graphically  two 
" simultaneous "  equations,  say,  x—2y=l,  and  2*+3y=16. 
The  lines  cross  each  other  at  the  point  P  (5,  2).  This  pair  of 
values  satisfies  both  equations  (let  class  verify).  A  line  drawn 
through  this  point  parallel  to  the  x  axis  is  y  =  2.  Hence 
the  value  of  x  for  both  lines  where  they  cross  y  =  2  is  5. 
The  5  represents  the  intercept  on  the  line  y  =  2,  made  by 
each  of  the  given  lines.  (Fig.  41.) 

2.  A  second  preliminary  principle  to  be  mastered  concerns 
the  method  of  making  out  tables  of  values  for  graphing. 
Having  decided  what  values  of  x  are  to  be  used  (this  is  a 
question  of  experience),  write  them  down  in  a  row,  then 


156 


CRAFTSMANSHIP  IN  MATHEMATICS 


evaluate  the  successive  parts  of  the  function,  one  complete 
row  at  a  time.  The  mental  work  proceeds  much  more  easily 
this  way  than  when  columns  are  completed  one  at  a  time. 
For  the  sake  of  comparison,  we  will  set  out  selected  values 


12 

Px 


X  -2i 

2oc  +  5i 

L 

Fig. 


1 

IG 


of  the  function  4#2  —  4#  —  Je^,  in  two  ways,  one  by  addition, 
one  by  multiplication.  Show  the  learner  why  the  results  are 
necessarily  identical. 


x= 

—  o 

36 
12 

-2 

16 

8 

-U 

-1 

4 
4 

0 

i 

1 
-2 

+  1 

4 
-4 

+  2 

+  2i 

+  3 

36 
-12 

+4 

64 
-16 

33 

i 

4r3-: 
-4z- 

9 
6 

0 
0 

16 

-8 

25 

-10 

y~4:Xz—4:X~  15  --  = 

33 

9 

0 

n 

-15 

-16 

-15 

-7 

0 

9 

Since  the  function  factorizes  into  (2x  +  3)  (2x  —  5),  we 
may  set  out  the  values  of  the  factors  and  multiply,  instead 
of  adding  as  before: 


x~ 

-3 

-2 

-li 

-1 

0 

i 

+  1 

+ 

+  2J 

+  3 

+4 

(2*+3)  = 
(2*-5)  = 

-3 
--11 

-1 

g 

0 

o 

+1 

-7 

3 
-5 

4 
_4 

5 
-3 

7 
-1 

8 
0 

9 
1 

11 
3 

y  =  4#*  —  4#  —  1  5  = 

33 

9 

0 

-7 

-16 

-16 

-16 

-7 

0 

9 

33 

GRAPHS 


157 


3.  A  third  preliminary  principle  concerns  scales.  Different 
scales  for  the  two  axes  are  often  desirable,  though  in  the  early 
stages  of  graphing  different  scales  are  not  advisable.  The 
learner  should  recognize  the  normal  slope  of  the  straight 
line  and  the  normal  shape  of  the  curve.  Only  in  this  way 
can  he  recognize  and  analyse  the  purely  geometrical  properties 
of  the  graph.  But  with  the  study  of  the  parabolic  function, 
if  not  before,  the  "  spread  "  of  the  numbers  should  be  taken 
into  account.  Moreover,  a  good  "  spread  "  to  the  parabola 
is  an  advantage,  in  order  to  obtain  accurate  readings  of  the 
x  intercepts. 

We  now  come  to  the  actual  graphing  of  the  function. 


—2,4 


-u-4 


2.4  C 


2     o 


-2 


-   X 


X 


Fig.  42 


Let  the  boy  be  first  made  familiar  with  the  graph  of 
the  normal  function  y  —  x2,  the  parabola  being  head  down 
and  the  co-ordinates  of  its  head  (vertex)  being  (0,  0).  Let 
him  see  that  the  curve  cuts  any  parallel  to  the  x  axis  in  two 
points,  e.g.  1  and  —1,  \/2  and  —  \/2,  &c.  The  curve  is 
symmetrical  with  respect  to  the  y  axis.  Note  that,  with  the 
same  scale  for  both  axes,  there  is  not  much  spread  to  the  curve. 

Now  we  will  graph  the  function  y  —  4#2  —  4^—15, 
taking  the  sets  of  values  for  x  and  y  from  either  of  the  tables 
on  the  previous  page.  To  obtain  a  greater  "  spread  ",  we 
adopt  a  larger  scale  for  the  x  axis.  The  curve  cuts  the  x  axis 
(when  .• .  y  =  0)  in  two  points,  viz.  — 1£  and  2£  (these  values 


158 


CRAFTSMANSHIP  IN  MATHEMATICS 


are  also  seen  in  the  tables),  and  these  are  therefore  the  roots 
of  the  equation  4#2  —  4#  — -  15  —  0. 

From  the  same  graph  we  may  obtain  the  roots  of  the 
equations  4#2  —  4x  —  15  =  9,  or  4#2  —  4#  —  15  —  —7,  or 
4#2  —  4#  —  15  =  zy  where  z  =  any  number  whatsoever.  It 
is  simply  a  question  of  drawing  across  the  curve  a  parallel 


V 


-3 


\ 


33 


\ 


-2 


ttLfi) 


£•* 


30 


10 


+  2 


/v 


K33) 


Fig.  43 


to  the  #  axis,  and  of  reading  the  values  of  x  from  the  points 
of  intersection.  For  instance,  if  4#2  —  4#  —  15  =  9,  the 
parallel  to  be  drawn  is  x  =  9,  and  this  cuts  the  curve  in 
x  =  —  2  and  3,  which  are  therefore  the  roots  of  the  equation. 
These  values  of  x  may,  of  course,  be  seen  from  our  tables 
where  y  —  4#2  —  4#  —  15  =  9,  but  they  are  easily  estimated 
from  the  graph  itself,  if  this  is  reasonably  accurate. 

A  function  may  sometimes  be  conveniently  divided  into 
two  parts,  and  each  part  treated  as  a  separate  function  and 


GRAPHS 


159 


graphed.  The  intersection  of  the  two  graphs  will  then  give 
the  roots  of  the  equation.  Really  we  have  two  simultaneous 
equations;  e.g. 

if         4*a  -  4*  -  15  =  0, 

then      4*2  =  4*  +  15. 


Hence  we  may  write 
and 


y  =  4#2 

y  =  4*  +  15 


The  line  cuts  the  curve  at  the  points  x  =  — 1£  and  2£,  and 
these  are  the  roots  of  the  equation  4#2  —  4#  —  15  =  0,  as 


10- 


**£ 


*0 


-ocr 


-2  -I 


-  4x-M5 

Fig.  44 


before.  It  should  be  noticed  that  this  last  figure  does  not 
represent  the  graph  of  the  function  y  =  4#2  —  4#  —  15, 
though  this  graph  is  now  easily  drawn  by  superposing  the 
4*2  graph  on  the  4*  +  15  graph.  If  Yt  =  4*2,  Y2  =  4^:  +  15, 
and  Y  =  4^2  -  4*  —  15,  then  Y  =  Yx  -  Y2.  Hence  any 
ordinate  of  Y  may  be  obtained  by  taking  the  algebraic  difference 
of  the  corresponding  ordinates  of  Y!  and  Y2.  Let  the  pupils 
draw  the  Y  graph  from  their  Yx  and  Y2  graphs,  and  verify. 


160 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  function  might  have  been  broken  up  in  another  way 

If  4xa  -  4*  -  15  =    0, 

then  4#a  —  4*  —  15. 

Hence  we  may  write  y  =  4jc2  —  4# 

and  y  =  15. 


(-3,48) 


\ 

/ 

K48) 

\ 

/ 

\ 

, 

y 

\ 

rJ 

\ 

(-• 

'^' 

\ 

i 

b,2 

> 

\ 

If 

=  if 

/ 

/ 

\ 

^ 

/ 

\ 

«,l 

tf 

( 

l,8> 

\ 

^ 

/•  — 

'2,8) 

^ 

15) 

S 

X 

/ 

5-2-10 

(^ 

1)   1           2           3           4^ 

Fig.  45 


Here  are  the  graphs  of  these  two  functions.  The  latter  cuts 
the  former  at  #—  —  1^  and  2J,  the  same  roots  as  before. 
The  easiest  way  to  discover  where  the  parabola  4#2—  4#— 15 
crosses  the  x  axis  is  to  express  the  quadratic  function  as  a 
product  of  two  linear  functions,  viz.  (2#  +  3)  (2#  —  5)  =  0. 
Hence  either  2#  +  3  =  0  or  2#— 5  =  0,  i.e.  x=  —  f  or  -ff. 
Thus  from  the  two  linear  functions  we  form  two  simple 
equations,  the  roots  of  which  are  the  roots  of  the  quadratic 
equation. 


GRAPHS 


161 


We  will  plot  these  two  linear  functions  (see  the  second 
table,  p.  156).  (The  lines  happen  to  be  parallel.  Why?) 
The  graph  of  the  quadratic  function  is  readily  obtained  by 
multiplying  together  corresponding  y  values  (again  refer  to 
second  table,  p.  156).  For  instance,  at  —2  the  y  value  of 
2x  +  3  is  —  1  and  the  y  value  of  2x  —  5  is  —9.  The  product 
of  —1  and  —9  is  +9.  Hence  at  —-2  the  y  value  of  the 
quadratic  function  is  +9,  i.e.  the  point  (—2,  +9)  is  a  point 


Fig.  46 

on  the  curve.    By  pursuing  this  plan  we  may  obtain  fig.  43 
over  again. 

The  boy  ought  now  to  realize  that  he  may  graph  his 
function  in  a  variety  of  ways.  But  do  not  encourage  him  to 
think  that  the  normal  process  of  solving  a  quadratic  equation 
is  to  graph  the  function.  Not  at  all.  The  important  thing 
for  the  boy  to  understand  is  that  every  algebraic  function 
can  be  thrown  into  a  picture  and  that  this  picture  tells  a 
story.  What  the  algebra  means  to  the  geometry  and  what 
the  geometry  means  to  the  algebra  are  the  things  that  matter. 
We  are  dealing  with  the  same  thing,  though  in  two  different 
ways,  and  the  closeness  of  the  relationship  should  be  seen 
clearly.  As  with  the  linear  function,  so  with  the  parabolic 

(B291)  I* 


162 


CRAFTSMANSHIP  IN  MATHEMATICS 


function:  the  boy  must  see  the  result  of  "  pushing  the  graph 
about  ". 


If 

then 
or 


—  4x  —  15  =  y, 
2  -  4*  +  1  =  y  +  16, 
(2x  -  I)2  =  y  +  16. 


If  we  compare  this  with  the  normal  form  x2  =  y>  we  see  that: 

2x  —  I    has  taken  the  place  of  x 
and  y  +  16  has  taken  the  place  of  yt 

i.e.  instead  of  x  =  0,    2x  —  I  =  0,    or  x  =  £, 

and  instead  of  y  =  0,    y  -f  16  =  0,    or  y  =  —16, 

i.e.  the  head  of  the  parabola  is  not  (0,  0)  but  (J,  —16)  as  in 
fig.   43.      Clearly  the  graph  of  4#2  —  4#  —  15  is  identical 


-  oc 


»  3oc 


GRAPHS  163 

with  the  graph  of  4#2,  except  that  it  has  been  pushed  J  unit 
to  the  right,  and  16  units  down.  (The  scale  difference  must, 
of  course,  be  borne  in  mind.) 

This  identification  of  similar  functions  is  of  great  im- 
portance throughout  the  whole  range  of  algebra.  One  of 
the  greatest  difficulties  of  beginners  is  to  see  how  the  form 
of  a  normal  function  may  be  obscured  by  mere  intercept 
values. 

Family  of  parabolas. — Let  the  boy  graph  a  few  related 
parabolas  like  the  following:  y  =  x2\  y  =  2#2;  y  —  3#2;  &c. 
For  2#2,  the  ordinates  of  x2  are  doubled;  for  3#2,  tripled; 
and  so  on.  Grouping  of  this  kind  helps  to  impress  on  the 
learner's  mind  the  relationship  of  the  curves. 

A  metal  rod  bent  into  the  shape  of  a  parabola,  with  an  in- 
conspicuous cross-piece  for  maintaining  its  shape  and  for  mov- 
ing it  about  the  blackboard,  is  useful  for  oral  work  in  class. 

Contrast  the  parabola  y  —  ax2  +  bx  +  c  when  a  is 
negative  with  that  when  a  is  positive.  With  a  negative,  the 
curve  is  "  head  up  ";  e.g.  7  +  3#  —  4#2  gives  such  a  parabola. 
Fig.  49  shows  another.  Give  the  boys  a  little  practice  in 
drawing  parabolas  in  this  position.  They  should  also  draw 
one  or  two  of  the  type  x  =  y2  and  x  =  —y2,  and  carefully 
note  the  positions  with  respect  to  the  axis. 

Turning -Points.     Maximum  and  Minimum  Values 

The  pupil  has  learnt  that  in  y  =  4#2  —  4#  —  15,  the  head 
of  the  parabola  is  (J,  —16).  He  sees  that  the  equation 
4#2  —  4#  —  15  —  0  has  two  roots  whenever  y  is  greater  than 
—  16.  For  example  if  y  =  9,  x  =  —  2  and  3;  if  y  =  —7,  the 
roots  are  —1  and  2;  if  y  =  —15,  the  roots  are  0  and  +1. 
But  if  y  =  —16  the  two  roots  are  equal,  each  being  *5.  '  If  a 
line  parallel  to  the  x  axis  is  down  below  y  =  16,  it  does  not 
cut  the  curve  at  all,  so  that  if  y  is  less  than  — 16,  x  has  no 
values,  or,  as  is  generally  said,  "  the  equation  has  no  roots  ". 
For  instance,  if  we  give  y  the  value  —17,  and  work  out  the 
equation  4#2  —  4#  —  15  =  — 17  in  the  ordinary  way,  we 


164 


CRAFTSMANSHIP  IN  MATHEMATICS 


find  that  x  = 


±V-i 


But  these  values  of  x  have 


no  reality  because  we  cannot  have  the  square  root  of  a  negative 
number.  The  graph  tells  the  true  story.  Instead  of  saying 
that  the  equation  has  two  unreal  or  "  imaginary  "  roots, 
we  may  more  correctly  say  that,  when  the  value  of  y  is  less 
than  —16,  x  has  no  value  at  all,  simply  because  the  y  line 


"  -7 


Fig.  48 

does  not  now  cut  the  curve  at  all.    The  y  line  is  "  out  of  the 
picture  ". 

As  a  point  moves  along  the  curve  from  the  left  downwards, 
the  ordinate  of  the  point  decreases  until  it  reaches  the  value 
-—16,  then  a  turn  upwards  is  made,  and  the  ordinate  begins 
to  increase  as  it  ascends  to  the  right.  The  point  (+-5,  — 16) 
is  the  turning-point  of  the  graph,  and  the  value  —16  of  the 
ordinate  is  called  the  turning  value  of  the  ordinate  (or  of  the 
function).  That  value  of  the  ordinate  is  its  minimum  value. 
If  the  graph  was  one  with  its  head  upwards,  the  turning- 
point  would  be  at  the  top  and  would  be  a  maximum  value 
(see  fig.  49). 


GRAPHS 


165 


Thus  the  pupil  must  understand  clearly  that,  in  the  case 
of  any  parabolic  function,  the  head  (vertex)  represents  a 
kind  of  limiting  value  of  y.  Each  value  of  y  corresponds  to 
two  different  values  of  x,  though  the  head  of  the  curve  seems 
to  be  an  exception.  Strictly  speaking,  the  head  corresponds 
to  only  one  value  of  #,  but  it  is  convenient  to  adopt  the  con- 
vention that  x  has  in  this  case  two  identical  values.  Beyond 
the  head,  outside  the  curve,  x  can  have  no  values.  Some 
quadratic  equations  have  two  roots,  some  OPP  (two  identical), 
some  none.  Do  not  talk  of  "  imagi.iary  '  roots:  that  is 
nonsensical.  We  shall  refer  to  this  point  again,  in  the  chapter 
on  complex  numbers  (see  Chap.  XXVII). 

The  pupil  should  note  how  slowly  the  length  of  the 
ordinate  changes  near  the  turning-point  of  a  parabola.  In 
fact  this  characteristic  of  slow  change  near  a  turning-point 
is  characteristic  of  turning-points  in  all  ordinary  graphs. 
Let  the  pupil  plot  on  a  fairly  large  scale  y  =  x2  for  small 
values  of  x. 


0 
•no 

x-  — 

^x. 

p 

I 

2  / 

? 

4 

^s 
5 

X 

7 

0 

-10 

/ 

^ 

\ 

-to 

\ 

/ 

s 

\ 

-V 

f 

\ 

1 

\ 

<-(2x2-!9oc+35) 

Fig-  49 


Show  the  pupil  how  the  graph  tells  him  at  a  glance 
where  the  values  of  y  (the  function)  are  positive,  say  for 
y  =  19x  —  2x2  —  35.  The  part  of  the  curve  above  the  x  axis 


166  CRAFTSMANSHIP  IN  MATHEMATICS 

corresponds  to  values  of  x  between  2£  and  7.  But  the  values  of 
y  (=  19#  —  2x2  —  35)  above  the  x  axis  are  positive.  Hence 
the  expression  19#  —  2x2  —  35  is  positive  between  the  values 
2£  and  7.  If  any  values  outside  these  are  tested  algebraically, 
the  expression  is  seen  to  be  negative.  (Fig.  49.) 

It  may  be  emphasized  again  that  quadratic  equations 
should  be  looked  upon  as  merely  one  interesting  and  useful 
feature  in  the  general  elementary  theory  of  parabolic  functions. 
Do  not  forget  practical  applications  of  the  parabolic  function; 
e.g.  falling  bodies  in  mechanics. 


Simultaneous  Equations 

Practice  in  solving  various  types  of  simultaneous  equations 
should  be  given  less  with  the  idea  of  finding  the  actual  roots 
of  the  equations  than  for  the  purpose  of  studying  the  relative 
positions  and  the  intersections  of  the  graphs.  We  will  refer 
briefly  to  two  typical  examples. 

1.    Consider  the  equations: 

xz  +  y*  =  97  \       and     x*  +  y*  =  20 


=  97  \ 
=  36  / 


xy  =  36          and  xy  =  36 

#2  +  y2  =  97  is  a  circle  with  its  centre  at  the  origin  and 
radius  A/97;  and  xy  —  30  is  a  rectangular  hyperbola  symmetri- 
cally placed  in  the  first  and  third  quadrants,  with  its  vertices 
at  a  distance  of  V2  X  36  from  the  origin.  As  V2  X  36  is 
less  than  V97,  the  circle  cuts  the  hyperbola  in  four  points, 
symmetrically  placed.  In  the  second  case,  since  V20  is  less 
than  V2  X  36,  the  circle  does  not  cut  the  hyperbola,  and 
there  are  no  roots.  (Fig.  50.) 
2.  Consider  the  equations: 


X  =  \r  — 


Here  we  have  two  parabolas,  one  with  its  apex  downwards, 
touching  the  axis  of  x,  two  units  to  the  right  of  the  origin^ 


GRAPHS 


167 


d 

, 

—y 

\ 

^ 

\^ 

\ 

/ 

/* 

\ 

s^ 

^ 

/ 

X 

/ 

\ 

/ 

/ 

\ 

o 

1 

A 

\ 

O 

/ 

\s 

S 

/ 

-^ 

\ 

^~ 

^ 

" 

A 

s 

\ 

3C%  o/  -  97 
ocy    =36 
OV  =A/^36;  OA  -  V97 

x'H-^-20 

OC'U'  -  3G 
OA  -  ~/ZO  ;  OV-  v'Z  x  l 

Fig.  50 


the  other  symmetrically  astride  the  x  axis,  with  its  apex  at  —1 
to  the  left.  The  roots  are  readily  obtained  approximately  by 
measurement  of  the  co-ordinates  of  the  intersections.  (Fig.  51.) 

jc 


4   - 


3    (- 


fo.O 


(c5r 

-2 


(20) 


-2) 


-  .(  X  -  2) 


Fig.  51 


168 


CRAFTSMANSHIP  IN  MATHEMATICS 


Higher  Equations 

The  pupils  should  study  a  few  cubics  graphically,  if  only 
that  they  may  gain  confidence  in  a  method  of  general  applica- 
tion. 

The  normal  form  of  the  cubic  (y  =  #3)  is  easily  graphed 
and  remembered. 


-  re 

Fig.  52 

Consider  the  equation  S(x  —  1)  (x  —  2)  (x  —  4)  =  0. 
Let  8(*  -  1)  (x  -  2)  (x  -  4)  -  y. 


*  = 

0 

1 

U 

2 

3 

4 

5 

(*-!)  = 

-1 

0 

i 

1 

2 

3 

4 

(*  -  2)  = 

2 

—  1 

i 

2 

0 

1 

2 

3 

(*  -  4)  = 

4. 

-3 

-21 

-2 

j 

0 

1 

y  =  8(*  -  1)  («  -  2)  (*  -  4)  = 

-64 

0 

5 

0 

-16 

0 

96 

The  curve  cuts  the  x  axis  at  points  1,  2,  and  4,  which  are 
therefore  the  roots  of  the  equation  (as,  of  course,  we  know 
at  once  from  the  factors).  (Fig.  53.) 


GRAPHS 


169 


(5  -16) 


^  =  8(x-l)(x-2)(x-4) 
Fig.  53 

Now  consider  the  equation  #3  —  7#  +  4  =  0. 

Since  *»  —  7*  +  4  =  0;    /.  *3  -=  lx  —  4. 

Let  ^  =  YI;  7*  -  4  =  Y2;  Y3  -  Yt  -  Y2  -  ^3  -  lx  +  4. 

We  will  tabulate  values  for  Ya,  Y2,  and  Y3. 


2 

-2 

-1 

0 

+  1 

+  2 

+  3 

Y,   = 

x3   = 

-27 

-8 

j 

0 

+  1 

+  8 

+  27 

Y2  = 

lx  -  4   = 

-25 

-18 

-11 

-4 

+  3 

+  10 

+  17 

Y3  - 

^3  -  7»  -i-  4   = 

-2 

-1-10 

+  10 

+  4 

-2 

2 

+  10 

We  will  now  plot  Yx  (=  #3),  a  normal  cubic,  and  Y2 
(=  7#  —  4)  a  straight  line,  and  so  solve  the  equation.  The 
latter  cuts  the  former  in  three  points,  viz.  where  x  =  —2-90, 
•60,  2'29,  which  are  therefore  the  three  roots.  But  the 


170 


CRAFTSMANSHIP  IN  MATHEMATICS 


figure  (fig.  54)  does  not  show  the  graph  of  y  =  of  — -  7x  +  4, 
the  original  function.  To  draw  this  graph,  we  may  either  use 
the  values  of  Y3  in  the  table,  or  superimpose  the  above  two 
graphs,  Yj  and  Y2,  on  each  other,  remembering  that 


7 


OC 

7oc-4 

Fig.  54 


Y3  =  Yt  —  Y2,  and  that  therefore  we  may  obtain  any  ordinate 
for  Y3  by  taking  the  difference  of  the  corresponding  ordinates 
for  Y!  and  Y2.  For  instance,  the  ordinate  at  x  =  2  is  —8 
for  Y!  -  a»,  and  -18  for  Y2  =  7x  -  4,  and  for  Y3(=  Yj-Yg) 
is  therefore  — -8+  18,  or  +  10.  And  so  generally.  This 


y 


.  55 


time  the  roots  of  the  equations  are  given  by  the  intersection 
of  the  curve  with  the  x  axis,  the  values  (—2-90,  -60,  2-29) 
being,  of  course,  the  same  as  before.  (Fig.  55.) 


172 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  Logarithmic  Curve 

We  dealt  with  the  A  B  C  of  Logarithms  in  Chapter  XI, 
and  we  now  come  to  the  logarithmic  curve,  the  use  of  which 
is,  of  course,  not  as  a  substitute  for  the  tables  but  as  a  justifi- 
cation of  the  extension  of  the  laws  of  indices  from  positive 
integers  to  fractional  and  negative  values.  The  boy  has  to 
learn,  too,  that  the  curve  is  really  a  picture  of  a  small  set  of 
tables.  He  should  therefore  be  taught  to  plot  a  curve  from 
first  principles,  and  to  use  it  as  far  as  he  can. 

Let  him  first  become  familiar  with  the  general  form 
of  the  curve.  For  instance  he  might  plot  y  =  2X,  3*,  5*. 


Go 
50 
40 
30 
20 
IO 


OC 


1 

1 

1 

/ 

x 

' 

—  ' 

s 

-  2 


l*-*2 


Ex 

FiR.  56 


3      4 


-  3 

4-8l 


T 


Ex.  5  - 125 


Show  the  advantage  of  changes  of  scales.  Draw  two  or  three 
extended  logarithmic  curves  on  the  blackboard,  and  spend 
a  few  minutes  in  oral  work,  e.g.  28?  36?  54?  (approximate 
answers  are  of  course,  all  that  can  be  expected). 

The  next  step  is  to  deal  with  the  evaluation  of  fractional 
indices  in  y  —  10*.  Let  the  class  graph  y  —  10*  up  to  x  =  3, 
on  a  fairly  large  scale,  drawing  the  graph  from  the  integral 
values  x  =  1,  2,  3.  "  If  the  index  law  holds  good,  we  ought 
to  be  able  to  obtain  by  readings  from  the  graph  such  values 
as  IO1*  and  IO2*.  But  our  graph  is  necessarily  very  rough; 


GRAPHS 


173 


we  had  such  a  few  points  with  which  to  plot  it.     We  must 
try  to  construct  a  better  curve. 

"  Let  us  use  our  arithmetic  for  constructing  the  curve, 
say  a  curve  representing  values  from  10°  to  101.  The  more 
values  we  find,  the  more  points  we  shall  have  for  plotting 
our  curve.  How  many?  Say  7  between  10°  and  101,  viz. 

10s  10*,  10&,  10*,  10s   10s  10&." 


Begin  with  10*  =  10  >  = 
Then  10*  -  10'.  = 
Then  10s  = 


3-162. 
1-779. 
=  1-333. 

We  have  4  more  to  find,  viz. 

10s  10«,  10s  101. 

I0l  =  (10<)3  =  (1-333)3  -  2-371. 

10§  ==  (10<)>  -  (1-333)-'  =  (1-333)2  X  (1-333)3 
-  1-779  X  2-371  =  4-217. 

I0l  =  10?  -  (101)3  -  (1-779)3  =  5-623. 

105  =  (lO1^)7  -  (1-333)7  -  (1-333)4  x  (1-333)3 
=  3-162  x  2-371  =  7-497. 

If  the  arithmetic  is  distributed  amongst  the  class,  it  is  quickly 
done;  very  little  explanation  is  necessary,  provided  previous 
elementary  work  in  powers  and  roots  was  understood. 

Now  the  boys  can  make  up  their  table  of  values,  changing 
the  vulgar  fractions  into  decimal  fractions;  then  plot  their 
points,  and  draw  the  curve. 


X  = 

0 

•125 

•25 

•375 

•5 

•625 

•75 
5-623 

•875 

1 

10 

y  =  10*  = 

1 

1-333 

1-779 

2-371 

3-162 

4-217 

7-497 

The  class  may  now  be  given  a  few  multiplication  and  division 
sums  to  work,  for  the  purpose  of  checking  their  curve.  (Of 
course  they  cannot  read  to  more  than  2  places  of  decimals.) 


174 


CRAFTSMANSHIP  IN  MATHEMATICS 


1.  Multiply  3-79  x  2-38. 

From  the  graph,  3-79  =  10'6'9  and  2-38  =  10'876. 

/.  3-79  x  2-38  =  10'"»  x  10-"«  =  10'9«  =  9-02  (from  the  graph). 

Now  verify  by  actual  multiplication. 


*90 


JE 
3 


«*£ 

422  — 


i2S~ 

2L_ 

1-78  — 


E%- 


10 


^5" 


UC 


7&: 


_LJ-A 


^Q*55. 


•02 


X' 


625 


ROWERS     TO    BASE  10. 
Fit?.  57 

2.  Divide  9-02  by  2-38. 
9-02  +  2-38 

=    10-9"   .i.    XO-876  =    JQ.955-370 

Now  verify  by  actual  division 


Now  let  the  class  write  into  their  graph,  by  interpolation, 
the  index  values  of  the  integral  numbers  1  to  10.     (Some 


GRAPHS 


175 


teachers  make  the  boys  learn  off  these  values  to  3  places  of 
decimals.)  The  boys'  interpolations  resulting  from  their 
own  measurements  will  necessarily  be  very  rough  and  at 
this  stage  a  prepared  graph  of  the  following  kind  might  be 
given  them. 


X  = 

0 

•301 

•477 

•602 

•699 

•778 

•845 

•903 

•954 

1-00 

y  =  10*  = 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

/ 

9 

-10 

954 

1 

/ 

ft 

=  10 

903 

/ 

/ 

7 

-10 

•645 

/ 

/I 

G 

-10 

•776 

./ 

'      \ 

r 

5 

-10 

•699 

/ 

/^ 

4 

-10 

602 

/ 

/^ 

3 

-10 

477 

, 

X 

/> 

^ 

2 

-10 

301 

^ 

<^ 

•^ 

^ 

,  

-*" 

i 

•50 

fA 

•6C 

-69 

t 

77JS 

\ 

•an 

$  •» 

3  *9( 

t 

, 

• 

•i 

'4 

•t 

f 

$ 

C 

> 

V 

•i 

i 

1 

POWERS   TO  BASE  I0f> 
Fig.  58 


The  term  "  logarithm  "  may  now  be  introduced.     "  It 
is  just  another  name  for  index."      Set  out  a  multiplication 


176  CRAFTSMANSHIP  IN  MATHEMATICS 

sum  in  parallel,  showing  the  related  methods.  Emphasize 
the  fact  that  the  two  things  are  the  same,  except  in  appear- 
ance. 

Multiply  4-73  by  1-84. 


1. 

Let  4-73  x  1-84  =  x. 
x  =  (4-73  X  1-84) 
==  10  fl76  X  10  265  (graph) 


=  IO940; 
/.  x  =  8-70  (graph). 


2. 

Let  4-73  x  1-84  =  x. 

log  x  =  log  (4-73  X  1-84) 
=  log  4-73  -f  log  1-84 
=  -075  -f  -265  (graph) 
=  -940; 
.'.  x  =  8-70  (graph). 


Now  give  the  boys  just  one  page  of  4-figure  logarithms,  make 
them  work  out  a  few  examples  in  both  ways,  and  see  they 
understand  that  the  two  ways  represent  exactly  the  same  thing. 
It  ought  now  to  be  possible  for  the  boys  to  proceed  with 
logarithms  in  the  usual  way,  and  really  to  understand  what 
they  are  doing. 

Graphs  and  the  "  Method  of  Differences  " 

The  nature  of  a  graph  may  easily  be  investigated  by  means 
of  the  method  of  differences.  A  series  of  equidistant  ordinates 
is  drawn,  beginning  at  any  point  on  the  graph.  The  heights 
of  the  ordinates  are  measured,  and  a  table  is  made  of  the 
first,  second,  third,  .  .  .  differences.  If  the  graph  is  a  straight 
line,  the  first  difference  will  be  constant;  if  a  parabola,  the 
second  difference;  if  a  function  of  the  third  degree,  the  third 
difference;  and  so  on.  Hence  by  examining  the  differences  of 
the  ordinates,  we  can  determine  the  degree  of  the  function 
which  corresponds  to  the  graph.  This  is  a  useful  principle 
for  the  boys  to  know. 

Books  to  consult: 

1.  Graph  Book,  Durell  and  Siddons. 

2.  Graphs,  Gibson. 


ALGEBRAIC  MANIPULATION  177 

CHAPTER  XVIII 

Algebraic    Manipulation 

Common -form  Factors 

During  the  last  30  years  there  has  been  amongst  the  older 
boys  of  schools  a  serious  falling  off  in  their  power  of  algebraic 
manipulation.  Nowadays,  there  is  often  a  sad  lack  of  easy 
familiarity  with  even  the  simpler  transformations  in  algebra 
and  trigonometry.  Although  a  great  deal  of  bookwork  is 
done  and  mastered,  the  valuable  old  transformation  exercises 
receive  too  little  attention,  with  the  result  that  there  is  often 
a  good  deal  of  uncertainty  about  everyday  working  algebraic 
procedure. 

Readiness  in  manipulation  is  the  key  to  algebraic  success. 
Pupils  must  acquire  facility  in  the  manipulation  of  common 
algebraic  expressions. 

The  factors  to  be  mastered  in  the  first  year  of  algebra  are 
few,  but  they  are  of  fundamental  importance  and  must  be 
taught  thoroughly.  In  the  early  stages  they  should  be  associated 
with  arithmetic  and  geometry,  if  only  in  order  that  the 
pupils  may  be  convinced  of  their  usefulness. 

The  early  forms  are, 

ab  ±  ac  =  a(b  ±  c), 

and  a2  -  b2  =  (a  +  b)  (a  -  6); 

and  the  expansions       (a  +  b)z  =  a2  +  2ab  -f  b2 
and  perhaps  (a  ±  &)3  -  a3  +  3a*b  -f  3ab2  +  fc3. 

Let  factors  be  first  looked  upon  as  a  device  for  simplifying 
formulae,  and  for  putting  these  into  shape  for  arithmetical  sub- 
stitution. It  is  a  good  plan  to  begin  with  obvious  geometrical 
relations  and  base  upon  these  an  algebraic  identity.  But  do 
not  talk  of  "  proving  "  the  truth  of  the  geometrical  pro- 
position. The  illustrations  in  Chapter  XVI,  pp.  134-7 
typify  the  kind  of  thing  to  be  done. 

(B291)  13 


178 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  elementary  standard  forms  (a  +  b)2,  (a  —  A)2, 
a2  —  b2y  being  well  known,  verified  by  a  few  numerical 
examples,  and  illustrated  geometrically,  a  first  element  of 
complexity  may  be  introduced  into  them. 

The  a  and  the  b  may  be  regarded,  respectively,  as,  say, 
a  square  and  a  circular  box,  into  each  of  which  we  may  put 
any  algebraic  expression  we  please.  Thus  we  may  write: 

D2-  O2  =  (D  +  0)(D  -  O), 
and  then  fill  up,  say  with  p2  and  q2  respectively,  in  this  way: 


.e. 


-  tf   =  (p* 


Q)  (/>- 


Such  a  device  is  very  useful,  but  do  not  carry  such  an  ex- 
tension very  far  at  first.  Wait  a  year,  and  then  with  harder 
examples  push  the  principle  home. 

The  expansions  (a  ±  b)3  are  probably  best  postponed 
until  the  second  year,  though  when  they  are  taken  up  they 
should  be  associated  with  a  geometrical  model.  A  6-in.  or  8-in. 
cubical  block,  sawn  through  by  cuts  parallel  to  each  pair  of 
parallel  faces,  makes  a  suitable  model,  and  may  be  prepared 
in  the  manual  instruction  room.  Or  a  cube  cut  from  a  bar 
of  soap  may  be  used,  if  a  very  thin-bladed  knife  is  available 
for  cutting  the  sections.  We  deal  first  with  the  identity 
(a  +  b)3  =  a3  +  3a2b  +  3ab2  +  b3.  If  each  edge  of  the 
cube  is  cut  into  two  parts  a  and  i,  the  original  edge  being 
a  +  b,  and  a  being  >  6,  the  cut-up  cube  evidently  consists 
of  eight  portions,  viz.  a  larger  cube  #3,  three  square  slabs 
of  area  a2  and  thickness  b,  three  square  prisms  of  length 
a  and  square  section  i2,  and  a  smaller  cube  b3. 

Then  the  class  sees  at  once  that 

(a  +  6)8  =  a8  +  3a*b  +  3a&8  +  b*. 


ALGEBRAIC  MANIPULATION 


179 


But  they  should  discover  this  identity  from  the  model  for 
themselves,  and  not  be  told. 

The  same  model  may  be  used  for  the  identity 


but  the  manipulation  is  a  little  more  troublesome.  The 
whole  composite  cube  must  now  be  called  #3,  and  the  thick- 
ness of  the  other  seven  parts  (slabs,  prisms,  and  small  cube) 
should  be  called  b.  The  larger  of  the  two  cubes  within  the 
whole  composite  block  is  evidently  (a  —  A)3.  When  actually 
handling  the  model  it  is  easy  to  see  that  this  cube  (a  —  6)3 
with  the  three  slabs  (3a2b)  and  the  little  cube  (63)  are  together 
equal  to  the  whole  composite  block  (#3)  plus  the  three  prisms 
(3«62),  i.e. 

(a  -  b)^  +  3a*b  +  63  -  <z3  -f  3ab\ 
or         (a  -  b)^  -  «3  -  Sa*b  -f  Sab*  -  b*. 

The  three  slabs  "  overlap  ",  a  fact  which  tends  to  perplex 
most  pupils. 

It  is  really  better  to  cut  up   two  cubes  and  have  two 
models,  one  to  be  kept  in  its  eight  separate  pieces,  the  other 


Fig.  59 


to  be  glued  up  again  without  the  cube  (a  —  i)3  and  looking 
something  like  three  of  the  six  sides  of  a  cubical  box.  Unless 
the  teacher  is  pretty  deft  in  manipulating  such  a  model,  it 
had  better  not  be  used,  or  the  class  will  get  more  amusement 


180  CRAFTSMANSHIP  IN  MATHEMATICS 

than  instruction  from  his  efforts.  It  is  obvious  that  since 
a  —  AB,  the  second  model  (the  three-sided  shell)  is  less  than 
the  three  slabs  3a26  by  the  three  prisms  Sab2  diminished  by 
the  little  cube  i3. 

I.e.         shell  -  3a2b  -  (3ab2  -  63) 


add  the  removed  cube  (a  —  6)3  to  each  side: 

shell  +  (a  -  6)3  -  (a  -  b)*  +  3a*b  -  3ab2  +  b*, 
i.e.  a3  =  (a  -  b)*  +  3a26  -  3ab*  -f-  63, 

i.e.         (a  -  b)*  -  a3  -  3a26  +  3a62  -  63  (as  before). 

This  on  paper  looks  complicated.  With  the  model  in  the  hand 
it  may  be  made  clear  at  once.  The  case  seems  complicated 
because  what  we  have  called  a  slab  a2b  consists  of  four  pieces 
of  wood,  each  of  the  thickness  b,  viz.  a  slab  (a  —  b)2  in  area, 
two  square  prisms  each  (a  —  b)  long,  and  a  cube  i3. 

The  boys  always  look  upon  it  as  a  pretty  little  puzzle. 
Let  them  build  up  the  cube  a3  themselves,  beginning  with 
the  cube  (a  —  6)3,  and  adding  and  subtracting  the  other 
pieces  one  by  one.  The  whole  difficulty  comes  about  from 
calling  the  edge  of  the  whole  cube  a  as  compared  with  the 
previous  example  when  a  referred  to  part  of  the  edge. 

A  further  identity  for  the  boys  to  discover  from  their 
model  is: 


The  whole  cube  may  be  called  a3  and  the  removable  cube 
i3.  Lay  out  the  seven  pieces,  all  of  thickness  (a  —  6),  on  the 
table.  The  united  area  obviously  is: 

3ab  +  (a  -  6)a 
=  a2  +  ab  +  b2; 

:.     volume  =  (a  —  b)  (a2  +  ab  +  b*)9 
i.e.        a3  -  &3  =  (a  -  b)  (a2  -f  ab  +  62). 

Verify  all  these  identities  by  a  variety  of  numerical  calculations, 
and  so  emphasize  the  utility  of  the  alternative  forms. 

It  is  a  curious  fact  that  Form  IV  boys  are  prone  to  forget 


ALGEBRAIC  MANIPULATION  181 

the  factors  of  a4  -f  a?b2  +  A4.  It  is  a  good  thing  to  ask  them 
occasionally  for  the  factors  of  (aQ  —  A6).  They  will  give 
them  readily  enough: 

=  (a8  +  63)  (a3  -  &3) 

=  (a  +  b)  (a2  -  ab  +  b2)  (a  -  b)  (a2  +  ab  +  b2). 

Now  ask  them  to  multiply  the  four  factors  together  again, 
in  pairs: 

(a  -f  b)  (a  —  b)  —  a2  —  b2  (readily  given), 

(a2  —  ab  +  b2)  (a2  +  ab  +  b2)  (generally  forgotten). 

If  the  product  is  not  forthcoming,  ask  for  the  factors  of 
#4  +  a2b2  -|~  £4  and  give  them  the  hint  of  adding  and  sub- 
tracting a262,  thus: 

a*  +  a2b2  +  64 
=  («4  -f  2a2b2  +  64)  -  a2b* 

-  (a2  -f  b2)2  -  W 

-  (a2  +  ab  +  b2)  (a2  -  ab  +  b2). 

Come  back  to  this  twice  a  term,  until  it  is  known. 


Algebraic  Phraseology 

Each  successive  school  year  will  demand  its  quota  of 
further  manipulative  work  until  in  the  Upper  Fifth,  especially 
the  top  Set,  the  boys  become  expert.  The  four  or  five  years' 
course  of  instruction  must  be  organized  in  such  a  way  that 
the  difficulties  of  manipulation  are  carefully  graded.  Im- 
press on  the  boys  that  ready  manipulation  is  the  key  to  success 
in  the  greater  part  of  algebra  and  therefore  to  the  greater 
part  of  trigonometry,  conies,  and  the  calculus. 

Let  your  phraseology  be  accurate,  and  use  it  consistently, 
exercise  after  exercise,  lesson  after  lesson,  and  see  that  the 
boys  gradually  acquire  the  use  of  phraseology  of  the  same 
degree  of  accuracy. 

"  Jones,  what  is  the  first  thing  to  do?" — "  Rearrange  the 
terms." 


182  CRAFTSMANSHIP  IN  MATHEMATICS 

"  How?"  —  "  Write  down  all  the  plus  terms  first,  and  then 
all  the  minus  terms." 

"  Then?"—  "  Put  the  plus  sign  .  .  ." 

"  No.  That  is  not  the  way  we  decided  to  say  things."  — 
"  Add  up  all  the  plus  terms  and  write  down  the  sum,  pre- 
fixed by  a  plus  sign;  then  add  up  all  the  minus  terms  and 
write  down  the  sum,  prefixed  by  a  minus  sign." 

"  Smith:  lastly?"  —  "  Take  the  difference  between  the 
two  sums,  and  prefix  the  sign  of  the  larger." 

Remember  the  slow  boys  and  the  amount  of  practice  they 
need  until  the  soaking  in  is  complete.  Then  all  is  well. 

There  are  certain  common  algebraic  terms  which,  though 
of  fundamental  importance,  are  often  loosely  used.  Formal 
definitions  to  be  learnt  by  rote  are  unnecessary,  but  con- 
sistently accurate  usage  should  be  adopted  from  the  outset. 
Introduce  the  terms  one  at  a  time  and  make  each  new  one 
part  of  the  everyday  jargon  of  each  lesson  for  a  few  weeks. 
We  refer  to  such  terms  as  mononomial,  binomial,  degree  and 
dimensions  ,  homogeneity  and  symmetry,  and  so  forth. 

"  In  algebra,  a  letter,  or  a  product  of  two  or  more  letters, 
or  of  letters  and  numbers,  in  which  there  is  no  addition  or 
subtraction,  is  called  a  term,  or  a  mononomial,  e.g.  #,  #2,  x2y, 


"  If  the  same  letter  occurs  more  than  once  in  a  term 
we  write  the  letter  down  once,  and  at  the  top  right-hand 
corner  we  write  a  figure  to  show  the  number  of  times  it 

occurs,  e.g.  xxx  is  written  #3,  aaaa  is  written  04. 

i  q 

"  A  term  may  be  integral^  as  ab2;  or  fractional,  as  —  . 

x 

"  The  degree  or  the  dimension  of  a  term  is  the  sum  of  the 
indices  of  the  named  letters;  e.g.  the  term  x2y3  is  a  term  of 
the  fifth  degree,  or  a  term  of  five  dimensions. 

"  A  binomial  consists  of  two  terms  connected  by  the 
sign  +  -or  —  ;  a  trinomial  of  three  terms;  a  polynomial  of 
more  than  three." 

All  this  is  just  the  stock  phraseology  of  the  classroom.  But 
let  it  be  carefully  thought  out  and  consistently  used,  in  order 


ALGEBRAIC  MANIPULATION  183 

that  the  boy  may  soon  get  to  know  the  precise  significance 
of  the  new  vocabulary. 

We  have  already  referred  to  the  term  function.  Use  it 
consistently  and  use  it  often. 

Such  a  term  as  the  law  of  commutation  is  hardly  worth 
mentioning  at  all  unless  it  be  in  Form  VI,  where  algebraic 
theory  is  being  minutely  discussed.  The  boys  will  know 
from  their  arithmetic  that  the  mere  order  in  which  terms  are 
arranged  for  addition  purposes  is  immaterial.  So  with 
multiplication:  the  notion  of  commutation  is  imbibed  with 
the  multiplication  table;  5  sevens  gives  the  same  product  as 
7  fives.  Thus,  any  elaborate  formal  explanation  that 
d+c  +  a  +  b  —  a  -{-  b  -{-  c  -{-  d,  or  that  b2ac  is  the  same 
as  ab2c,  is  unnecessary.  It  is,  as  a  rule,  enough  to  point  out 
the  close  analogy  with  arithmetic,  though  in  a  first-year  course 
of  algebra  attention  must  repeatedly  be  called  to  the  fact 
that  abc  is  not  in  form  a  faithful  copy  of  345,  and  that  345 
means  300  +  40  +  £>.  In  the  main,  let  early  algebraic 
processes  grow  out  of  corresponding  arithmetical  processes. 


Typical  Expressions  for  Factor  Resolution 

1.  ac  +  be  -f  ad  +  bd  =  (a  +  b)  (c  +  d). 

2.  x2  +  (a  +  b)x  +  ab  =  (x  +  a)  (x  +  6). 

3.  acx*  +  (ad  +  bc)x  +  bd  =  (ax  +  b)  (ex  +  d). 

These  depend  on  a  redistribution  of  terms,  and  too  much 
care  cannot  be  paid  to  the  teaching  of  the  principle  involved. 
We  know  that 

(a  +  b)(c  +  d)  =  a(c  +  d)  +  b(c  +  d) 
=  ac  +  ad  +  be  +  bd, 

and  therefore,  conversely, 

ac  +  ad  +  be  +  bd  =  a(c  +  d)  +  b(c  +  d) 

d). 


If  then  we  are  given  the  expression  ac  +  bd  +  ad  +  be, 


184  CRAFTSMANSHIP  IN  MATHEMATICS 

and  we  rearrange  it  so  that  both  the  a  terms  come  first,  we 
have  a  suitable  distribution  for  finding  the  factors: 

ac  +  bd  +  ad  -f-  be 
=  ac  +  ad  -f-  be  +  bd 
=  a(c  +  d)  +  b(c  +  d) 
=  (a  +  b)(c  +  d). 

Boys  are  often  puzzled  about  the  derivation  of  the  last 
line  from  the  last  line  but  one,  but  their  difficulty  is  cleared 
up  when  it  is  pointed  out  to  them  that  if  they  had  to  multiply 
(a  +  b)  by  (c  +  d),  they  would  begin  by  writing  down 
a(c  +  d)  +  b(c  +  d). 

Emphasis  must  be  laid  on  this  intermediate  step  of  a 
partial  redistribution  and  on  how  we  proceed  forwards  and 
backwards  from  it. 

(a  -f  b  -\-  c)  (d  +  e) 
=  a(d  +  e)  -f  b(d  +  e)  f  c(d  +  e) 
=  ad  -\-  ae  -j-  bd  -f-  fee  +  cd  +  £<?, 
which  /.      =  a(rf  +  r)  -f  6(rf  +  e)  +  c(rf   f-  e) 

—  (a  +  fe  +  c)  (d  +  e),  with  which  we  began. 

We  append  two  rather  harder  examples.  It  is  always  a 
question  of  arranging  according  to  the  powers  of  some 
selected  letter,  though  which  letter  only  experience  can  tell. 

(i)  x*  +  (a  +  b  +  c)x  +  ab  +  ac 

—  x2  -f  ax  -f  bx  +  £#  +  fl&  +  #£• 

Arranging  in  powers  of  a,  we  have 

ax  +  0fe  +  ac  +  #a  +  &#  +  £# 
=  ^(A;  +  b  +  c)  +  x(x  +  b  +  c) 
=  (a  +  x)  (x  +  6  +  c). 

(ii)  a2  -H  206  -  2ac  -  362  -f  2bc. 

We  note  the  letter  c  in  two  terms.  Try  grouping  them 
together. 

Then  a2  +  2ab  -  3&2  -  2ac  +  2fo 

=  (a2  +  206  -  362)  -  2c(a  -  b) 
=  (a  +  36)  (a  -  b)  -  2c(a  -  b) 
=  (a  +  36  -  2c)  (a  -  6). 


ALGEBRAIC  MANIPULATION  185 

If  boys  feel  a  difficulty  about  accepting  the  last  line  as  another 
form  of  the  last  line  but  one,  give  them  an  example  of  the 
reverse  kind: 

(a  +  b)  (c  +  d  +  e) 

either      =  (a  +  b)c  +  (a  +  b)d  +  (a  +  b)e 

or  =  (a  +  b)  (c  +  d)  +  (a  +  b)e. 

Both  redistributions  yield  exactly  the  same  result. 
Illustrate  with  a  numerical  example: 

47  x  365 

either       =  (47  x  300)  +  (47  x  60)  +  (47  x  5) 
or         =  (47  X  360)  -f  (47  X  5), 

i.e.  we  can  perform  our  multiplication  in  little  bits  or  in 
bigger  bits,  just  as  we  please. 

The  type  x2  +  (a  +  b)x  +  ab  =  (x  +  a)  (x  +  b)  seldom 
gives  much  trouble.  Examples: 

x2  +  Sx  +  15. 
x*  -  8*  +  15. 

The  two  rules  (1)  for  signs,  (2)  for  determining  the  coefficients 
of  x,  should  be  kept  separate.  Both  admit  of  very  simple 
statement. 

For  the  first  example  we  begin  by  writing  (x  +  )  (#  +  )> 
and  for  the  second  example  we  begin  by  writing  (x  —  )  (x  —  ). 
For  both  examples  we  ask  the  question,  What  two  numbers 
multiplied  together  give  us  15  and  when  added  together 
give  us  8?  Answer,  5  and  3.  Hence  the  factors  (x  +  5)  (x  +  3) 
and  (x  —  5)  (x  —  3). 

Other  examples: 

*a  +  2x  -  15 
x2  -  2x  -  15. 

As  the  last  term  is  a  minus  term,  the  second  term  of  the 
two  factors  will  be  of  opposite  signs.  Hence  we  may  begin 
by  writing  down  for  each  case  (x  +  )  (x  —  ).  "  Find  two 
numbers  whose  product  is  15  and  whose  difference  is  2." 


186  CRAFTSMANSHIP  IN  MATHEMATICS 

Answer,  5  and  3.  "  Give  the  larger  number  the  sign  before 
the  middle  term."  Hence  we  have: 

x*  +  2x  -  15  =  (x  +  5)  (x  -  3) 
x*  -  2x  -  15  =  (x  -  5)  (x  +  3). 

Of  course  these  are  mere  rules,  to  be  remembered;  but  they 
should  be  first  worked  out  from  an  examination  of  the  different 
products,  three  or  four  sets  being  taken  for  confirmation 
purposes. 

(x  +  3)  (x  +  5)  =  x2  +  8*  +  15 

(x  -  3)  (x  -  5)  =  x2  -  Sx  +  15 

(x  +  5)  (*  -  3)  =  x*  +  2*  -  15 

(x  -  5)  (x  -f  3)  =  x*  -  2x  -  15. 

Help  the  boys  to  examine  the  products  and  to  discover: 

(1)  That  if  the  last  term  of  the  trinomial  is   +,  the 

signs  of  both  factors  are  the  same,  the  same  as 
the  middle  term. 

(2)  That  if  the  last  term  of  the  trinomial  is  — ,  the 

signs  of  the  two  factors  are  different,  the  factor 
with  the  larger  number  taking  the  sign  of  the 
middle  term. 

(3)  That  the  last  term  of  the  trinomial  is  always  the 

algebraic  product  of  the  second  terms  of  the 
two  factors  (hence  the  signs). 

(4)  That  the  middle  term  of  the  trinomial  is  always 

the  algebraic  sum  of  the  second  terms  of  the 
two  factors  (hence  the  signs). 

The  mere  rules  must  be  mastered  by  all  Sets,  but  experience 
shows  that  the  justification  of  the  rules,  by  an  analysis  of  a 
series  of  products,  is  beyond  lower  Sets,  though  upper  Sets 
always  appreciate  them.  Do  not  talk  of  "  proving  "  the  rules. 

The  type,  acx2  +  (ad  +  bc)x  +  bd 

This   common  type   of  expression   boys   generally  find 
rather  troublesome  to  factorize.   I  remember  seeing  a  Fourth 


ALGEBRAIC  MANIPULATION  187 

Form  trying  to  factorize  35#2  —  59#  —  48.  There  had  been 
a  preliminary  discussion  on  the  necessarily  long  succession 
of  "  trial  "  factors,  and  the  33  boys  were  actually  working 
out  with  the  patience  of  33  Jobs  the  possible  combinations, 
the  first  factors  being  35*  ±  1,  7*  ±  1,  5#  ±  1,  #  ±  1, 
35*  ±  2,  7x  ±  2,  5*  ±  2,  x  ±  2,  and  so  on  with  ±3,  ±4, 
±6,  ±8,  ±12,  ±16,  ±24,  and  ±48,  80  possible  first  factors 
in  all!  Naturally  the  lesson  was  not  long  enough  for  this 
single  set  of  trials  to  be  completed.  In  any  circumstances 
the  particular  example  would  be  very  difficult  for  class 
practice.  But  the  "  trial  "  method  is  unnecessary.  All  ordinary 
cases  can  be  dealt  with  by  a  method  which  is  much  simpler. 
Consider  the  example  (6#2  +  17*  +  12)  =  (3#  +  4) 
(2x  -f-  3).  Let  us  multiply  the  factors  together  in  the  ordinary 
way. 

3*  +  4 

2x  H-_3 

ftica~~+~8jc 

4-  9*    +12 


+  12 


We  might  have  multiplied  out,  thus: 

(3*  +  4)  (2*  +  3) 
=  2*(3*  +  4)  +  3(3*  +  4) 
=  6*2  +  8x  -f  9*  +  12 
=  6*2  +  17*  -f  12. 

To  find  the  factors,  why  not  reverse  this  process? 

6*2  +  17*  +  12 

=  6*a  +  8*  +  9*  +  12 
=  2*(3*  +  4)  +  3(3*  +  4) 
=  (3*  +  4)  (2*  -f  3). 

Yes,  why  not?  But  how  could  we  tell  that  the  ITx  in  the 
first  line  should  be  divided  into  8#  and  9#,  instead  of,  say, 
into  3#  and  14#,  or  into  5x  and  12#?  That  is  the  trouble, 
that  the  only  difficulty.  How  are  we  to  find  the  two  correct 
numbers? 


188  CRAFTSMANSHIP  IN  MATHEMATICS 

Let  us  suppose  these  unknown;   call  them  m  and  n. 

Now  m  +  n  =  17  (that  we  know). 

And  mn  =  72. 

[How  do  we  know  that?  Because  72  is  the  product  of  the 
6  and  12  which  we  obtained  (in  the  multiplication  sum)  by 
multiplying  3  by  2  and  by  multiplying  4  by  3;  and  from  these 
same  4  numbers,  3,  2,  4,  3  we  obtained  the  9  and  the  8  also 
in  the  multiplication  sum.  Thus  the  72  is  the  product  of  the 
6  and  12  in  the  first  and  third  terms  of  the  trinomial.] 

Hence  all  we  have  to  do  is  to  find  two  numbers  which 
when  added  together  come  to  17  and  which  when  multiplied 
together  come  to  72.  The  numbers  are  easily  seen  to  be 
8  and  9,  and  therefore  we  now  know  that  the  17*  must  be 
divided  into  Sx  and  9#. 

Another  example:    14#2  —  25*  +  6. 

Here  m  +  n  =  —25  and  mn  =  14  X  6  =  84.  By  trial, 
the  two  required  numbers  are  — 21  and  — 4. 

14*2  -  25*  +  6 
=  14*2  -  21*  -  4*  +  6 
=  7x(2x  -  3)  -  2(2*  -  3) 
=  (7*  -  2)  (2x  -  3). 

Another  example:   6#2  —  llx  —  10. 

This  time  we  have  to  find  two  numbers  whose  product 
is  — 60  and  whose  sum  is  — 11.  The  numbers  are  evidently 
-15  and  +4. 

6*2  -  11*  -  10 
=  G*2  -  15*  +  4*  -  10 
=  3*(2*  -  6)  +  2(2*  -  5) 
=  (3*  +  2)  (2*  -  5). 

Thus  we  have  this  simple  rule.  Redistribute  the  terms  of 
the  expression ,  splitting  the  coefficient  of  the  middle  term  into 
two  parts,  m  and  n,  so  that  m  and  n  is  the  product  of  the  co- 
efficients in  the  first  and  last  terms.  Then  factorize  the  re- 
distributed product  in  the  usual  way. 

For  top  Sets  the  rule  can  be  stated  more  formally  from 


ALGEBRAIC  MANIPULATION 


189 


the  expression  acx2  +  (fld  +  bc)x  +  bd,  where  the  relations 
stated  in  the  rule  are  obvious. 

At  least  top  Sets  should  be  made  to  see  how,  as  regards 
both  coefficients  and  signs,  all  the  different  cases  may  be 
brought  under  a  single  rule.  Let  the  general  expression  be 
axz  +  bx  +  c.  Then:  divide  b  into  two  parts  m  +  n  so 
that  win  =  ac.  Now  let  the  class  apply  the  rule  to  all  possible 
different  cases,  say: 

x2  ±  Sx  +  15, 

x2  ±  2x  -  15, 

6:c2  +  19*  +  15, 

6*2  +  x  -  15. 

Difficult  cases  where  m  and  n  cannot  be  obtained  readily 
from  mn  and  m  +  ;/  at  once  by  mental  arithmetic  may  be 
solved  quadratically.  Examples: 


x*  +  2x  -  360. 

Write,  x2  +  2x  -  360  --=  0. 
/.  x2  +  2x  +  1  =  361, 
/.    x  +  1  ==  ±19, 
.'.    x  =  18  or  —  20, 

/.  factors  =  (x  —  18)  (x  +  20). 


x2  +  12*  -  405. 

Write,  x*  +  12x  -  405  =  0. 
/.  x*  -f  I2x  +  36  =  441, 
.'.    x  +  6  -  +21, 
/.    x==  -27  or  +15, 

/.  factors  =  (x  -  15)  (x  +  27). 


Do  not  let  the  pupils  look  upon  these  as  quadratic  equations 
but  simply  as  a  plan  for  finding  the  factors.  Quadratic 
equations  will  come  a  little  later.  The  quadratic  principle 
may  be  applied  to  any  case,  but  more  often  than  not  it  is 
merely  a  clumsy  substitute  for  the  method  first  mentioned. 
For  instance,  consider  6x2  —  llx  —  10. 

6*2  _  11^  _  10  =  6(*2  -  Y*  -  |) 
Solving  the  quadratic  x2  —  ~^x  —  J j-  =  0,  we  have, 

#2  __  i  i  x  _|_  /i  i\2  _  »  _|_  /J.i\a 


or  —  f  . 


-  11*  -  10  =  6*  - 


f) 


=  (2*  -  5)  (3*  +  2). 


190  CRAFTSMANSHIP  IN  MATHEMATICS 

Complex  derivatives  from  type  forms  are  a  prolific  source 
of  errors  with  all  but  the  ablest  pupils.  Much  care  is  necessary 
in  substituting.  Example:  factorize  8a3  —  (a  +  2)3. 

Type:        x*  -  y*  =  (x  -  y)  (x*  +  xy  +  y2). 
Thus  x  =  2a;  y  =  a  +  2. 

(2a)3  -  (a  4-  2)3 

=  {2«  -  (a  +  2)}  {(2a)»  +  2a(a  +  2)  +  (a  +  2)2} 
-  (a  -  2)  (4a 2  +  2a2  4-  4a  +  a2  +  4a  +  4) 
=  (a  -  2)  (7a2  4-  8a  4-  4). 


Product  Distribution  Generally 

There  comes  a  time,  probably  towards  the  end  of  the 
Upper  Fourth  year  or  the  beginning  of  the  Lower  Fifth, 
when  a  boy's  accumulated  facts  concerning  products  must 
be  summarized  and  analysed,  and  reduced  to  laws  of  some 
kind.  We  will  run  rapidly  over  the  necessary  ground. 

(a  4-  b)  (c  4-  d).  Here  we  have  two  factors,  each  of  two 
terms.  We  have  to  multiply  each  term  of  the  first  factor  by 
c  and  then  by  d  and  so  we  have  four  terms  in  all,  viz. 

ac  4-  be  4-  ad  4-  bd. 

Compare  this  with  the  ordinary  arithmetical  multiplication. 

37  x  24 

=  (30  +  7)  (20  4-  4) 

=  (30  x  20)  4-  (30  x  4)  4-  (7  X  20)  4-  (7  X  4), 

and  show  the  close  analogy.     We  have  and  must  have  four 
products  both  in  the  algebra  and  in  the  arithmetic. 
For  similar  reasons: 

(i)  (a  4-  b  4-  c)  (d  4-  e)  will  give  6  products. 

(ii)  (a  4-  b  4-  c)  (d  +  e  +/)  will  give  9  products, 
(iii)  (a  4-  b)  (c  4-  d)  (e  +/)  will  give  8  products, 
(iv)  (a  +  b  +  c)  (d  4-  e  +/)  (g  +  h  +  k)  will  give  27 
products. 


ALGEBRAIC  MANIPULATION  191 

The  last  will  be  quite  clear  if  it  be  observed  that  each  of  the 
9  products  in  (ii)  has  to  be  multiplied  by  g,  then  by  h,  then 
by  k.  Clearly,  then,  if  the  factors  consist  of  p9  q,  and  r  terms, 
the  number  of  products  will  be  p  X  q  X  r\  and  this  will  be 
quite  general.  Hence  we  can  tell  how  many  products  to  expect 
in  an  algebraic  multiplication. 

But  in  the  above  cases,  all  the  terms  are  different.  There 
is  neither  condensation  owing  to  like  terms  occurring  more 
than  once,  nor  reduction  owing  to  terms  destroying  each 
other.  Either  or  both  of  these  things  may  happen. 

Consider  the  product  (a  -f-  b)  (a  -}-  b).  By  the  general 
rule  the  distribution  will  give  4  terms'  But  only  2  different 
letters,  a  and  b,  occur  in  the  product,  and  with  these  only  3 
really  distinct  products  of  2  factors  can  be  formed  with  them, 
viz.  a2,  aby  b2.  Hence,  among  the  4  terms,  at  least  1  must 
occur  more  than  once,  and,  in  fact,  a  X  b  occurs  twice. 
The  result  of  the  distribution  therefore  is  a2  +  2ab  +  b2. 

Thus  we  may  write: 

(a  +  6)2  =  a2  +  2ab  +  b*. 
Similarly      (a  -  6)2  =  a2  -  2ab  +  b2. 

In  the  case  of  (a  +  b  )(a  —  b),  the  term  ab  occurs  twice, 
but  as  the  two  terms  are  of  opposite  signs  they  destroy  each 
other.  Nevertheless  the  main  rule  still  holds  good:  the 
product  really  consists  of  2  X  2  or  4  terms. 

What  are  all  the  possible  products  of  3  factors  that  can 
be  made  with  the  2  letters  a  and  6?  Evidently 

aaa,       aab,       abb,        bbb\ 
or,       a3,  a26,          a&2,  63;    4  in  all. 

Hence  in  the  distribution  of  (a  +  6)3,  i.e.  of  (a  +  b)  (a  +  b) 
(a  +  b),  which  by  the  general  rule  will  give  8  terms,  only 
4  really  distinct  terms  can  appear.  What  terms  recur  and 
how  often? 

a3  and  i3  evidently  appear  each  only  once,  because  to 
get  3  a's  or  3  6's  we  must  take  one  from  each  bracket,  and 
this  can  be  done  in  only  one  way. 


192  CRAFTSMANSHIP  IN  MATHEMATICS 

cPb  may  be  obtained: 

(i)  by  taking  b  from  the  first  bracket,  and  a  from  each 

of  the  others; 
(ii)  by  taking  b  from  the  second  bracket,  and  a  from 

each  of  the  others; 
(iii)  by  taking  b  from  the  third  bracket,  and  a  from  each 

of  the  others. 

ab2:  the  same  holds  as  for  a2b. 
Thus  the  result  is, 


(a  +  &)3  =  «3  +  3a26  -f-  3ab*  +  63. 

Similarly,       (a  -  fc)3  -  a5  -  3a*b  +  3«62  -  b\ 

(a  +  &)4  =  <z4  - 


If  we  remember  that  the  possible  binary  products  of  3 
letters,  a,  b,  c,  are  a2,  i2,  c2,  ab,  acy  be  (6  in  all),  then 

(a  +  6  +  c)2  =  a2  +  62  +  c*  +  2ab  +  2ac  +  2bc. 

The  ternary  products  of  3  letters,  a,  b,  cy  are  easily  enu- 
merated if  we  first  deal  with  the  letter  a,  writing  down 
the  terms  in  which  it  occurs  thrice,  then  those  in  which  it 
occurs  twice,  then  those  in  which  it  occurs  once;  then  deal 
similarly  with  6,  for  such  forms  as  are  not  already  written 
down;  then  with  c.  Thus  we  have  (10  in  all): 

<23,  d*by  a2c,  ab2,  ac2,  abc, 

63,  b*c,  bc\ 

(*. 

Hence,  following  the  rule,  we  have: 

(a  +  b  +  c)3  -  (a  +  b  +  c)  (a  +  b  +  c)  (a  +  b  +  c) 

-f  3a2c  +  3ab2  -f 
Qabc. 


The  result  may  be  verified  by  successive  distribution: 

(a  +  b  +  c)3  =  (a  +  6  +  c)2  (a  +  b  +  c) 

=  (a2  +  ft8  +  c2  +  2ab  +  %ac  -f  2bc)  (a  +  b  -f-  c) 


ALGEBRAIC  MANIPULATION  193 

Another  example:   (b  +  c)  (c  +  a)  (0  +  b). 

Here  not  all  the  10  permissible  ternary  products  can  occur, 
for  a3,  63,  c3  are  excluded  by  the  nature  of  the  case,  a  appear- 
ing in  only  2  of  the  brackets,  b  in  only  2,  and  c  in  only  2. 

(b  +  c)  (c  +  a)  (a  +  6)  =  fo2  +  bzc  +  ca2  +  c2a  -f  <z&2  -f  a26  +  2afo. 

But  although  we  do  not  get  the  10  ternary  products,  we  do 
get  8  (—  2  x  2x  2)  products,  according  to  the  general  rule. 
In  the  product  (b  —  c)  (c  —  a)  (a  —  b)  the  term  abc 
occurs  twice  but  with  opposite  signs,  and  there  is  then  a 
further  reduction: 

(b  -  c)  (c  -  a)  (a  -  b)  =  be2  -  b*c  -f  caz  —  c*a  +  ab*  -  a*b. 


S  Notation 

Upper  Sets  in  the  Fifths  should  be  taught  to  use  this 
notation.  It  is  easy  to  understand,  though  average  boys 
are  a  little  shy  of  it.  "  S  "  stands  for,  the  sum  of  all  the  terms 
of  the  same  type  as,  though  its  exact  meaning  in  any  particular 
case  depends  on  the  number  of  variables  that  are  in  question. 

If  2  variables,  a  and  6,  ^Lab  means  simply  ab. 

If  4  variables,  #,  i,  c,  d,  %ab  means  ab  +  ac  +  ad 

+bc  +  bd  +  cd. 

If  2  variables,  a,  i,  2a26  means  a2b  -f-  ab2. 
If  3  variables,  a,  b,  c,  *Ldlb  means  a*b  +  ab2  +  a*c 

+  ac2  +  b2c  +  be2. 

The  context  usually  makes  clear  how  many  variables  are  to 
be  understood. 

"  Choose  any  one  of  the  terms  and  place  S  before  it." 
The  use  of  the  sign  certainly  saves  labour:   thus 


(a  +  &)3  =  Sa3 
(a  +  b  +  c)9  =  Sa3  +  3Sa26  +  Gabc. 
(b  +  c)  (c  4-  a)  (a  +  b)  =  Z.a*b  -f  2o6c. 

(E291)  14 


194  CRAFTSMANSHIP  IN  MATHEMATICS 

More  Complex  Forms 

Quick  boys  soon  pick  up  the  method  of  manipulating 
more  complex  forms  based  on  those  already  familiar  to 
them,  but  the  slower  boys  require  much  practice  and  should 
not  be  worried  by  forms  so  complicated  as  to  be  puzzling. 
The  slower  boys  should  always  first  be  given  forms  with  + 
signs  only.  The  added  difficulty  of  negative  signs  should 
come  a  little  later. 

(i)  Type  forms,  with  the  addition  of  coefficients;    e.g. 

(a)  (3a  +  2ft)3  -  (3«)3  +  3(3a)2  (2b)  +  3(30)  (2ft)2  -f  (2ft)3. 
(P)  (a  +  2ft  +  3c)2  =  &c. 

(ii)  Type    forms    with    a    mononomial    replaced    by    a 
binomial;    e.g. 

Replacing  b  throughout  by  (b  -f-  c)y  we  have 


(a  +  b  +  c)3  =  a3  +  3a*(b  +  c) 
=  &c. 

(iii)    Association    of   parts   of   factors   of   more    than    3 
terms;    e.g. 

(a)  (a  +  b  -f  c  -  d)  (a  -  b  +  c  +  <*) 

-  {(a  +  *)  +  (6  -  <0)  {(a  +  c)-(b-  d)} 

-  (a  -f  c?  -  (b-  d)\  &c. 

(P)  («  +  6  +  c)  (6  +  c  -  a)  (c  +  a  -  b)  (a  +  b  -  c) 

=  {(b  +  c)  +  a}  {(b  +  c)  -a}{a~(b-  c)}{a  +  (b  -  c)} 

-  {(b  +  c)2  -  a2}  (a2  -  (6  -  c)2} 

=  (62  +  2bc  +  c2  -  a2)  («2  -  fe2  -f  2bc  -  c2) 
=  {26c  +  (62  +  c2  -  a2)}  {2fo  -  (62  +  c*  -  a2)} 
=  (26c)2  -  (62  -f  c2  -  a2)2 


-  a4  -  fc4  -  c*. 


The  type  forms  are  few  and  are  easily  remembered,  and 
all  boys  should  have  them  at  their  finger-ends.  The  possible 
applications  and  developments  are,  of  course,  very  diverse, 


ALGEBRAIC  MANIPULATION 


195 


but  do  not  perplex  boys  with  expressions  that  are  beyond 
their  manipulative  skill  at  any  particular  stage. 


Detached  Coefficients.    First  Notions  of  Manipulation 

Here  is  a  useful  general  theorem,  easy  for  upper  Sets  to 
remember. — If  all  the  terms  of  all  the  factors  of  a  product  be 
simple  letters  unaccompanied  by  numerical  coefficients  and 
all  +,  the  sum  of  the  coefficients  in  the  distributed  value 
of  the  product  will  be  /  X  m  X  n  .  .  .  ,  where  /,  my  n,  are 
the  numbers  of  the  terms  of  the  respective  factors. 

Thus  in  the  evaluated  products  of  the  following,  we  have: 


Coefficients. 

Sum  of 
Coefficients. 

(a  +  b)* 

1  +  2+1 

4  =  22 

(a  +  b)3 

1+3+3+1 

8  =  23 

(a  +  &)« 

1  +  4  +  6  +  4  +  1 

16  =  2* 

(a  +  b)5 

1  +  5+10+10  +  5+1 

32  =  25 

(b  +  c)  (c  +  a)  (a  +  b) 

1  +  1  +  1  +  H-  1  +  1  +  2 

8  =  2s 

The  theorem  is  useful  in  connexion  with  expansions. 

Simple  Expansions  and  First  Generalizations  therefrom.  — 
Let  an  upper  Set  in  a  Fourth  Form  obtain  the  following 
results  by  continued  multiplication,  the  second  being  obtained 
by  multiplying  the  first  by  (x  +  #),  the  third  by  multiplying 
the  second  by  (x  +  a)y  and  so  on. 


(x  +  a)2  =  x2  +  2xa  +  a2. 
(x  +  a)3  =  x3  +  3*2a  +  3 
(x  +  a)*  =  x4  +  4x3a  + 
(x  +  a)6  -  x5 
(x  +  a)6  -  x6 


a3. 
+  4xa3  + 


15*2a4  +  Gxa*  +  a6. 

Now  help  the  boys  to  generalize,  and  to  establish  the  usual 
rules.  Afterwards,  they  may  work  out  a  few  higher  expansions 
and  see  that  these  follow  the  same  rules. 

1.  The  power  to  which  we  have  carried  (x  +  a)  gives 


196  CRAFTSMANSHIP  IN  MATHEMATICS 

the  index  of  the  highest  terms  of  the  expansion  and  is  there- 
fore the  degree  of  the  function. 

2.  The   function   has   one   term   more   than   that  index. 
Thus  the  expansion  (x  •+  #)4  has  5  terms. 

3.  The  powers  of  x  run  in  descending  order  from  the 
first  term  to  the  last  term  but  one;    the  powers  of  a  run  in 
ascending  order  from  the  second  term  to  the  last.    (There  is 
no  objection  to  writing  the  first  term  xna®  and  the  last  xQan> 
if  the  class  understand  that  x°  —  a°  =  1.    Then  both  x  and  a 
appear  in  every  term.) 

4.  The  dimensions  of  all  the  terms  are  the  same  and  are 
always  equal  to  the  power  to  which  (x  +  a)  is  carried. 

5.  The  coefficients  follow  a  regular  law. — We  may  detach 
them  from  their  terms  (detached  coefficients  may  often  be 
usefully  considered  alone),  and  place  them  in  order,  thus: 


(*  +  0)1 
(*  +  *)2 
(x  +  a)3 
(*  +  aY 
(x  +  a)5 
(x  +  a)* 

I         1 
1         2 

1 

1 

6 

1         3 

1        Y~ 
1         5 
1         6 

3         1 
6         4          1 

10       |10         5 
15       20~~~~|15 

The  sum  of  any  2  successive  coefficients  in  any  line  gives 
the  coefficient  standing  in  the  next  line  immediately  below 
the  second  of  these.  Thus,  in  the  third  line  the  6  is  the  sum 
of  3  and  3  in  the  second  line;  in  the  last  line,  the  15  is  the 
sum  of  10  and  5  in  the  fifth  line.  Show  that  this  is  the  simple 
result  of  continued  multiplication.  For  instance,  if  we  multiply 
(x  +  fl)3  by  (x  -f  a)  we  have: 

1 +3+3+1 

1  +  1 

1+3+3+1 

1+3+3+1 
1+4+6+4+1 

The  second  partial  product  is  arranged  one  place  to  the  right 
under  the  first  partial  product.   Thus  any  coefficient  for  any 


ALGEBRAIC  MANIPULATION  197 

expansion  may  be  found  by  taking  the  coefficient  of  the 
corresponding  term  in  the  previous  expansion  and  adding 
to  it  its  predecessor.  Let  the  boys  continue  the  table:  they 
like  the  work.  They  soon  see  that  when  they  have  written 
the  expansion  of,  say,  (x  -f-  a)10,  they  can  immediately  write 
down  that  for  (x  +  0)11;  it  is  merely  a  question  of  carrying 
on  the  game  already  begun. 

"  There  is  something  still  more  interesting  to  learn  about 
the  coefficients.  Consider  the  expansion  of  (x  +  of.  The 
coefficient  of  the  second  term  is  5;  we  may  write  it  {-.  The 
coefficient  of  the  next  term  is  10,  which  we  may  write  ^|. 
That  of  the  next  term  is  again  10,  which  we  may  write 
1x2x3*  ^nc*  so  genera^y-  Examine  the  other  expansions 
and  see  if  a  similar  rule  is  followed;  for  instance,  (x  +  a)Q 

6    *      ,    6.5    A  „    ,    6.5.4    ,  „    .    6.5.4.3    9  . 
5  *4 


/      ,      N« 
(x  +  a)6  = 


-  -----  __ 

1.4  L.^i.o  l.Z.o.4 

6.5.4.3.2      .        6.5.4.3.2.1    6 


With  one  or  two  leading  questions,  the  boys  will  see  that  the 
coefficient  of  a6  is  the  same  as  that  for  XQ,  that  for  xa5  the 
same  as  for  rr>a,  that  the  coefficient  —^  is  the  same  as 
c~^,  and  so  generally.  Let  them  formulate  the  obvious  rule 
for  themselves. 

Let  them  write  down  the  first  few  terms  of  such  an 
expansion  as  (x  +  fl)20- 

First  they  write  the  terms  without  coefficients: 

x20  +  xl9a  +  xl*a*  +  &c. 

Then  they  work  out  their  coefficients  and  insert  them: 
.    ,20  .   20.19  ,   Q 


The  object  of  all  this  is  not  to  teach  the  Binomial  Theorem: 
that  will  come  later.  It  is  to  impress  boys  with  the  wonderful 
simplicity  and  regularity  that  underlies  algebraic  mani- 


198  CRAFTSMANSHIP  IN  MATHEMATICS 

pulation.  Never  mind  the  generalized  form  (x  +  a)n,  yet. 
Never  mind  the  general  term.  Never  mind  nCr.  When 
these  things  are  taken  up  later,  the  way  will  have  been  paved, 
and  the  work  will  give  little  trouble. 


The  Remainder  Theorem 

This  theorem  must  be  known  in  order  that  the  Factor 
Theorem  may  be  clearly  understood.  It  may  be  approached  in 
this  way.  —  We  know  that  (x—  5)  is  a  factor  of  (x2-\-x  —  30), 
and  in  order  to  find  out  the  other  factor  we  may  conveniently 
set  out  the  process  of  division,  exactly  as  in  arithmetic. 

x  -  5)*2  +  x—  30(#  +  6 
x2  -  5* 

0*  —  30 
6*  -  30 


Of  course  there  is  no  "  remainder  "  (R),  but  if  (x  —  5)  was 
not  a  factor  there  would  be  a  R.  Divide  (3#2  —  2x  +  4) 
by  (x  -  5). 

x  -  5)3*2  -  2*  -f  4(3*   f    13 

??L_Z.  I5x 

13*+    4 

13*  -  65 

69  =  R. 

The  remainder  is  69,  and  by  analogy  with  arithmetic  we  know 
that 

Dividend  =  (Quotient  X  Divisor)  -f  R. 

Suppose  we  had  to  divide  (x2  +  px  +  q)  by  (x  —  a). 

x  —  a)x*  —  px  +  q(x  +  (a  —  p) 
x2  —  ax 

x(a  —  p)  +  q 

x(a  -  p)  -  a(a  -  p) 

a(a  —  p)  -f  q  =  R. 

Note  that  a(a  —  p)  +  q  really  is  the  remainder,  for  it  does 


ALGEBRAIC  MANIPULATION  199 

not  involve  x,  and  we  cannot  proceed  with  the  division  any 
farther. 

Now  let  us  set  out  in  this  way  the  previous  example, 
treating  the  figures  as  if  they  were  letters,  and  not  actually 
multiplying  and  subtracting  as  we  did  before. 

x  -  5)3x2  -  2x  +  4(3*  +  (3.5  -  2) 


x(3.5  -  2)  +  4 

x(3.5  -  2)  -  5(3.5  -  2) 

5(3.5  -  2)  +  4  =  R. 

As  might  be  expected,  the  R  is  the  same,  viz.  69. 

Now  compare  the  Remainders  and  the  Dividends  in  both 
the  last  examples. 

Dividend     =  x2  —  px  +  q.     I     Dividend     =  3#a  —  2x  +  4. 
Remainder  =  a2  —  pa  +  q.     \     Remainder  =  3.52  —  2.5  +  4. 

Clearly,  then,  if  the  remainder  was  the  only  thing  we  wanted, 
we  could  have  written  it  down  at  once,  for  it  is  exactly  of  the 
same  form  as  the  dividend.  We  merely  have  to  substitute 
for  the  x  in  the  dividend  the  second  term  of  the  divisor 
(a  in  the  first  case,  5  in  the  second),  treating  these,  however, 
as  if  they  were  positive. 

Give  the  pupils  several  examples,  and  convince  them  of 
the  truth  of  the  rule. 

The  Remainder  "  Theorem  ",  as  it  is  called,  provides  us 
with  a  simple  means  of  calculating  the  remainder  of  a  particular 
kind  of  division  sum  in  algebra,  without  actually  performing 
the  division. 

The  particular  kind  of  division  sum  is  that  in  which  the 
divisor  and  the  dividend  are  functions  of  the  same  letter 
(say  #),  and  in  which  the  divisor  is  a  linear  expression  such 
as  (x  —  5)  with  unity  as  the  coefficient  of  x. 

Example:   if  we  divide 

(*3  -  7*2  +  5*  -  1)  by  (x  -  9),  the  R  is 
(93  -  7.92  +  5.9  -  1)  =  206. 

We  have  merely  substituted  9  for  x  in  the  dividend. 


200  CRAFTSMANSHIP  IN  MATHEMATICS 

Hence: 

(1)  The   Theorem. — When  a  function  of  x  is  divided 
by   (x  —  a),   the   R   is   obtained   by   substituting 
a  for  x  in  the  function. 

(2)  The  why  of  it.    We  know  that, 

Dividend  =  (Quotient  X  Divisor)  -f  R. 

If  we  make  a  equal  to  x,  the  divisor  (x  —  a)  =  0. 
/.   Dividend  =  (Quotient  X  0)  -f  R, 

-R, 
i.e.  by  substituting  a  for  x  in  the  Dividend,  we  have  the  R. 

The  Factor  Theorem 

What  is  the  remainder  when  we  divide  (#2  —  7x  -f-  10) 
by  (x  -  5)? 

Substituting  5  for  x  in  x2  —  Ix  -|-  10, 

we  have  52  —  7.5  -\-  10 

-  0. 

Since   R  =  0,  (x  —  5)  divides   exactly  into  (x2  —  7x  +  10) 
and  is  therefore  a  factor  of  it. 

Thus  we  have  a  method  of  finding  out  whether  an  ex- 
pression of  the  type  (x  —  a)  is  a  factor  of  a  given  expression 

Example:  Is  (3*3  -  2*2  -  7*  -  2)  divisible  by  (x  -  2)  (x  +  1)? 

Writing  2  for  x  in  the  first  expression,  we  have 

24  —  8—14   -  2  -=  0.     Hence  (x  —  2)  is  a  factor, 

Again,  writing  —1  for  x  we  have 

-3-2  +  7-2-0.     Hence  (x  +  1)  is  a  factor. 

Homogeneous  Expressions 

A  homogeneous  expression  is  one  in  which  all  the  terms 
have  the  same  dimensions;   e.g. 

x*  +  xy  +  y*>      or      a3  +  63  +  c*  +  Sabc. 


ALGEBRAIC  MANIPULATION  201 

It  may  sometimes  be  necessary  to  state  what  letters  are  re- 
garded as  giving  dimensions;  e.g.  x3  +  ax2y  +  2xy2  +  3y3 
is  homogeneous  in  x  and  y  but  is  not  homogeneous  if  a  is 
regarded  as  having  dimensions. 

Obviously  the  product  of  two  homogeneous  expressions 
is  itself  homogeneous. 

The  only  homogeneous  integral  functions  of  x  and  y 
of  the  first  and  second  degrees  are, 

Ax  4-  By, 

Ax2  4-  "Rxy  +  Qy2. 

For  3  variables  the  corresponding  functions  are, 

Ax  -|-  I*y  -1-  C#, 

Ax2  4-  By2  4-  C*2  4-  D;vx?  +  Esx  +  Fxy. 

The  class  may  usefully  write  down  functions  of  the  third 
degree.  Upper  Form  boys  should  be  thoroughly  familiar 
with  all  such  general  expressions. 


Symmetry 

A  function  which  is  unaltered  by  the  interchange  of  any 
two  of  its  variables  is  said  to  be  symmetrical  with  regard 
to  these  variables',  e.g.  x2  —  xy  4-  y2  is  symmetrical  with 
regard  to  x  and  y]  (y  +  #)  (z  +  x)  (x  -\-  y)  is  symmetrical 
with  regard  to  x,  y,  and  z.  But  x2y  +  y2z  4-  ~2#  is  not  a 
symmetrical  function  of  x,  y,  and  #,  for  the  3  interchanges 
x  with  y,  y  with  z,  z  with  x,  give,  respectively, 

yzx  +  ff2*  +  #2}>, 
x2z  +  #2;y  4-  :V2#» 
z*y  4-  y2*  4-  A, 

and  although  all  these  are  equal  to  each  other,  none  of  them 
is  equal  to  the  original  expression. 

But  the  term  "  symmetry  "  is  not  used  in  quite  the 
same  sense  by  all  writers  in  algebra.  "  Cyclic  symmetry  " 
expresses  a  much  clearer  connotation. 


202  CRAFTSMANSHIP  IN  MATHEMATICS 

Cyclic  Expressions 

An  expression  is  said  to  be  "  cyclic  "  with  regard  to  the 
letters  ay  b,  c,  d,  .  .  .  k>  arranged  in  this  order,  when  it  is 
unaltered  by  changing  a  into  b,  b  into  c,  .  .  .  k  into  a.  Thus 
the  expression  a2b  +  b2c  +  c2d  +  d2a  is  cyclic  with  regard 
to  the  letters  a,  i,  £,  d,  arranged  in  this  order,  for  by  inter- 
changing a  into  b,  b  into  c,  c  into  d,  d  into  a,  we  get 
b2c  -f  c2d  +  d2a  +  a2b,  the  same  as  be- 
fore. Note  that  the  first  term  is  changed 
to  the  second,  the  second  to  the  third, 
and  so  on.  It  is  merely  a  question  of 
beginning  at  a  different  point  on  the 
circle,  but  always  going  round  in  the 
same  direction. 

Fig.  60  If  in  a  cyclic  expression  a  term  of 

some  particular  type  occurs,  the  terms 
which  can  be  derived  from  this  by  cyclic  interchange  must 
also  occur,  and  the  coefficients  of  these  terms  must  be  equal. 
Thus,  if  xy  y,  and  %  are  the  variables,  and  the  term  x2y 
occurs,  then  all  the  terms  x2yy  x2zy  y2z,  y2xy  %2x,  z*y  must 
occur.  The  cycle  is  easily  seen  if  the  six  terms  are  thus 
collected  up: 

x*(y  +  *)  +  y*(*  +  x)  +  z\x  +  y). 

Expressions  which  are  unaltered  by  a  cyclical  change  of  the 
letters  involved  in  them  are  called  cyclically  symmetrical. 
Thus  (b  —  c)  (c  —  a)  (a  —  b)  is  cyclically  symmetrical,  since 
it  is  unaltered  by  changing  a  into  i,  b  into  c,  and  c  into  ay 
that  is  "  by  starting  at  a  different  point  in  the  circle  ". 

Legitimate  Arguments  from  Cyclical  Symmetry 

Find  the  factors  of 

a\b  -  c)  +  b*(c  -a)  +  c\a  -  b).  .          .     .    (i) 

Here  is  the  solution  from  one  of  the  best  textbooks  we  have. 


ALGEBRAIC  MANIPULATION  203 

"  If  we  put  b  =  c  in  the  expression,  the  result  is  zero, 
and  it  therefore  follows  from  the  Remainder  theorem  that 
(b  —  c)  is  a  factor. 

"  In  a  similar  manner  we  can  prove  that  (c  —  a)  and 
(a  —  b)  are  factors. 

"  Now  the  given  expression  is  of  the  third  degree;  it 
can  therefore  have  only  3  factors. 

"  Hence  the  expression  is  equal  to 

N(6  -  c)  (c  -  a)  (a  -  6), (ii) 

where  N  is  some  number  which  is  always  the  same  for  all 
values  of  a,  ft,  c. 

"  We  can  find  N  by  giving  values  to  a,  ft,  c,  Thus,  let 
a  =  0,  ft  =  1,  <:  =  2;  then  (i)  =  -2,  and  (ii)  =  +2N. 
Hence  N  =  -I. 

"  Therefore  the  factors  are  —  (b  —  c)  (c  —  a)  (a  —  ft)." 

This  argument  is  open  to  criticism.  It  is  wholly  un- 
necessary to  say,  "  in  a  similar  manner  we  can  prove  that 
(c  —  a)  and  (a  —  ft)  are  factors".  Once  we  know  that  (ft  —  c) 
is  a  factor,  it  follows  at  once  that  (c  —  a)  and  (a  —  ft)  are 
factors.  What  applies  to  (ft  —  c)  must  apply  to  (c  —  a)  and 
(a  —  ft).  This  is  the  very  essence  of  cyclical  symmetry. 
Nay,  it  is  the  very  essence  of  all  algebraic  manipulation. 
That  (c  —  a)  and  (a  —  ft)  are  also  factors  requires  no  argu- 
ment of  any  sort  or  kind,  except,  "  it  follows  from  cyclical 
symmetry  ";  and  no  further  argument  should  be  tolerated. 

Another  example.  Find  the  factors  of  a3(b  —  c)  +  b\c  —  a) 
+  c3(a  —  ft).  As  in  the  last  example  (ft  —  c),  (c  —  #),  (a  —  ft) 
are  all  factors.  Now  the  given  expression  is  of  the  fourth 
dimension;  hence  there  must  be  a  fourth  factor,  and  that  of 
the  first  dimension.  Since  this  factor  must  be  symmetrical 
with  respect  to  #,  ft,  £,  it  is  necessarily  (a  +  ft  +  c).  Thus 
the  required  factors  are 

N(6  -c)(c-a)(a-b)(a  +  b  +  c), 

N  being  found  in  the  usual  way.  Any  sort  of  more  elaborate 
process  or  argument  should  be  sharply  criticized. 


204  CRAFTSMANSHIP  IN  MATHEMATICS 

Another    example.      Find    the    product    of   (a  +  b  +  c) 

(a*  _|_  #2  _j_  cz  _  fa  _  ca  __  ab^ 

Each  of  the  two  factors  is  symmetrical  in  a,  6,  c,  and 
therefore  the  product  will  be  symmetrical  in  a,  6,  £. 

Obviously  the  term  a3  occurs  with  the  coefficient  unity; 
hence  the  same  must  be  true  of  63  and  c3. 

Obviously,  too,  the  term  Ire  has  the  coefficient  0;  hence 
by  symmetry  the  five  other  terms  Ire,  c2ay  ca2,  ab2y  a2b  belong- 
ing to  the  same  group  must  have  the  coefficient  0. 

Lastly,  the  term  —  abc  is  obtained  (i)  by  taking  a  from 
the  first  bracket  and  —be  from  the  second;  hence  it  is  also 
obtained  (ii)  by  taking  6,  and  (iii)  by  taking  c,  from  the  first 
bracket.  Thus  the  term  abc  must  have  the  coefficient  —3. 
Hence  the  product 

=  a3  +  fe3  +  c3  -  3abc. 

Boys  should  gain  complete  confidence  in  arguments  from 
symmetry.  In  at  least  the  A  Sets  of  the  Fifth  Form,  cumbrous 
processes  should  be  prohibited  whenever  arguments  from 
symmetry  are  possible. 


Identities  to  be  Learnt 

The  following  identities  should  be  at  the  finger  ends  of 
all  Fifth  Form  boys. 

1.  (b-c)  +  (c-a)  +  (a-b)  =  0. 

2.  a(b  -  c)  -f  b(c  -  a)  -f  c(a  -  b)  =  0. 

3.  (a  +  b  -|-  c)2  =  a2  f  b2  +  c2  +  2a6  +  2bc  -f  2ca. 

4.  (a  +  b  -f  c)3  -  a3  4-  63  -f  c3  +  3b2c  +  3bc2  +  3c2a  +  3oia 

-f  3a2Z>    f-  M2  +  Gabc. 

5.  (a  +  b  +  c)  (a2  +  b2  +  c2  -  be  —  ca  —  ab)  =  a3  -f  ft3  -f-  & 

-Zabc. 

6.  (6  -  c)  (c  -  a)  (a  -  b)  =  -«2(6  -  r)  -  £2(c  -  a)  -  c2(a  —  b) 

=  —  fo(6  —  c)  —  ra(c  —  a)  —  ab(a  —  b). 
1.  (b  +  c)  (c  +  a)  («  +  6)  -  a»(6  +  c)  +  b2(c  +  a)  +  c2(a  +  6) 


ALGEBRAIC  MANIPULATION  205 

and  perhaps, 

8.  (a  +  b  +  c)  (a2  +  62  +  c2) 

=  bc(b  +  c)  4  ca(c  4  a)  +  ab(a  +  b)  +  a3  +  b3  4  c3. 

9.  (a  +  b  4-  c)  (be  4  ca  4  aft) 

=  02(6  4  c)  4  ft2(c  +  a)  4-  £2(a  +  ft)  4  3«fo. 
10.  (a  4  ft  4  c)  (ft  4  c  -  a)  (c  4  a  -  ft)  (a  4  6  -  c) 
=  2ft2c2  4  2c2a2  4  2a2ft2  -  a4  -  ft4  -  c4. 

Books  to  consult: 

1.  Textbook  of  Algebra  (2  vols.),  Chrystal  (still  the  leading  work 

in  the  subject). 

2.  A  New  Algebra,  Barnard  and  Child. 


CHAPTER  XIX 

Algebraic   Equations 

Equations  of  Different  Degrees 

"  Either  .  .  .  or?" 

More  than  once  I  have  heard  a  small  boy  round  on  his 
teacher  for  this  kind  of  argument: 

Solve  the  equation  x2  —  7x  +  12  =  0.  "  Factorizing,  we 
have  (x  —  4)  (x  —  3)  =  0.  Hence  either  (x  —  4)  or  (x  —  3) 
must  be  zero,  i.e.  x  must  be  either  4  or  3,  and  therefore 
both  4  and  3  must  be  roots  of  the  equation." 

Says  the  boy:  "  You  said  either  (x  —  4)  or  (x  —  3)  must 
be  zero;  how  then  can  it  follow  that  x  is  both  4  and  3?" 

The  criticism  is  just,  for  the  reasoning  is  faulty. 

A  formal  approach  to  equations  may  be  successfully 
made  by  such  general  arguments  as  follows. 

It  is  advisable  in  the  first  place  to  distinguish  between 
an  equation  and  an  identity,  and  consistently  to  use  the  same 
form  of  words  when  referring  to  them.  For  instance:  "When 
two  expressions  are  equal  for  all  the  values  of  the  quantities 


206  CRAFTSMANSHIP  IN  MATHEMATICS 

involved,  the  statement  of  their  equality  is  called  an  identity" 
e.g.  that  m  — •  (n  —  p)  =  m  —  n  -{-  p  is  true  for  all  values  of 
the  letters  m,  ny  and  p. 

"  But  when  two  expressions  are  equal  for  only  particular 
values  of  the  quantities  involved,  the  statement  of  their 
equality  is  called  an  equation."  Thus  #  +  7  =  10  is  an 
equation;  it  is  true  only  where  x  —  3. 

If  in  an  equation  we  bring  all  the  terms  from  the  right- 
hand  side  to  the  left-hand  side,  and  equate  the  whole  to  0, 

e-8-  x  +  7  -  10  =  0, 

then  by  giving  x  its  own  particular  value,  the  expression 
"  vanishes  ",  e.g. 

3  +  7  _  10  =  0, 

i.e.  3  +  7  —  10  is  seen  really  to  be  0,  and  has  therefore 
"  vanished  ". 

The  value  of  the  unknown  quantity  that  makes  the  two 
sides  of  an  equation  equal  is  said  to  satisfy  the  equation. 
The  process  of  finding  that  value,  the  root,  is  called  solving 
the  equation. 

Consider  the  equation  3(#  —  2)  =  2(x  —  1). 

For  what  value  of  x  is  3(x  —  2)  equal  to  (2x  —  1)? 

Try  a  few  values,  say  the  numbers  1  to  10.  The  only  one 
of  the  ten  that  makes  the  expressions  equal  is  4,  i.e. 

3(4  -  2)  =  2(4  -  1), 

and  so  we  say  that  4  is  the  root  of  the  equation. 
If  we  simplify  the  original  equation,  we  have 

3*  -  6  =  2x  -  2, 
/.   3*  -  2x  =  6  —  2, 

.".  x  —  4,  as  expected. 

We  may  write  #=4  as  x  —  4  =^  0,  and  when  in  the  ex- 
pression x  —  4  we  write  4  for  x,  the  expression  vanishes, 
for  4  -  4  =  0. 

Again,  for  what  value  does  x2  — -  x  =  6? 

Try  a  few  numbers  as  before.    We  find  that  in  this  case 


ALGEBRAIC  EQUATIONS  207 

there  are  two  values  which  satisfy  the  equation,  viz.  3  and 
—2,  and  that  there  are  no  others.  Substituting,  we  have 

32  -  3  -  6, 
and  (-2)2  -  (-2)  =  6. 

If  we  write  the  equation  in  the  form  x2  —  x  —  6  =  0,  the 
expression  on  the  left-hand  side  vanishes  when  we  write 
either  x  =  3  or  x  =  —2.  Thus 

32  -  3  -  6  =  0, 
and          (-2)2  -  (-2)  -  6  -  0, 

and  it  does  not  vanish  for  any  other  value. 

With  equations  of  the  second  degree,  we  may  always 
find  two  values  of  x  that  will  satisfy  the  equation. 

Since  x2  —  x  —  6  =  0, 

and  since  x*  —  x  —  6  =  (x  —  3)  (x  -f-  2), 

.'.  (x  -  3)  (*  +  2)  =  0. 

Now  a  product  cannot  be  equal  to  zero  unless  one  of  the 
factors  is  equal  to  zero,  and  hence  (x  —  3)  (x  +2)  can  be 
equal  to  zero  only  (1)  when  x  —  3  —  0,  and  (2)  when  x  +  2 
=  0,  and  never  otherwise.  Thus  when  we  have  (x  —  3) 
(x  +  2)  =  0,  we  may  equate  the  two  factors  equal  to  0 
separately,  solve  the  two  simple  equations,  and  obtain  the 
two  roots: 

Since          x  —  3  =  0,     .*.  x  =  3, 
and  since         x  -f  2  =  0,      .*.  x  =  —  2;  as  before. 

Thus  we  have  a  method  of  solving  a  quadratic  equation. 
Bring  all  the  terms  to  the  left-hand  side  and  equate  to  0; 
break  up  the  expression  into  factors,  and  equate  each  of  these 
to  0;  solve  the  two  resulting  simple  equations. 

Note  that  a  quadratic  equation  has  two  roots. 

Here  is  a  quotation  from  a  well-known  textbook:  "  If 
the  product  of  two  quantities  is  nothing,  one  of  the  quantities 
is  nothing."  One  objection  to  this  phraseology  is  that  "  noth- 
ing "  is  not  a  mathematical  term. 

Suppose  we  have  an  equation  of  the  third  degree,  a 


208  CRAFTSMANSHIP  IN  MATHEMATICS 

cubic  equation  as  it  is  called,  say,  x*  —  6#2  —  llx  =  6. 
Bring  all  the  terms  to  the  left-hand  side  and  equate  to  0. 
By  a  series  of  trials  we  may  discover  that  there  are  3  and 
only  3  values  of  x  which  will  make  the  expression  on  the  left- 
hand  side  vanish,  viz.  1,  2,  and  3  Thus 

#3  -  6*2  I-  11*  --0  =  0 
I3  —  6(l)2  +  11  —  6  =  0 
23  _  6(2)2  +  22-6-0 
33  _  6(3)2  4-33-6-0. 


Hence  the  roots  of  the  equation  are  1  ,  2,  3.  But  the  trials 
would  have  been  tedious.  Let  us  factorize  as  before. 

Since          x*  -  6*2  +  11*  -  6  ==  0, 
(x  -!)(*-  2)  (*  -  3)  =  0. 

A  product  of  factors  can  be  equal  to  0  only  when  one 
of  its  terms  is  equal  to  0.  Obviously,  in  the  equation,  this 
may  happen  in  three  different  ways,  when  (x  —  1)  =  0,  when 
(x  —  2)  =  0,  when  (x  —  3)  --  0,  and  in  no  other  way.  If 
then  we  solve  these  three  simple  equations,  we  get  x  =  1 
or  2  or  3,  as  before.  A  cubic  equation  has  three  roots. 

Suppose  we  have  an  equation  of  the  fourth  degree,  a 
"  biquadratic  "  equation  as  it  is  called.  It  is  quite  easy  to 
solve  if  we  can  factorize  the  expression  made  by  bringing 
all  the  terms  to  the  left-hand  side.  Usually  this  is  a  difficult 
job,  but  here  is  an  easy  one. 

x*  +  Qx*  -f  38*  =  8jc3  +  40, 
/.  #*  -  8*3  +  9*a  +  38*  -  40  -  0, 
/.  (x  -!)(*  +  2)  (x  -  4)  (x  -  5)  -  0. 

This  product  can  be  zero  only  if  one  of  its  factors  is  zero. 
This  can  happen  in  4  ways,  and  only  in  4,  viz.  when  x  —  1  ==  0, 
#-j-2  =  0,#  —  4  =  0,  x  —  5  =  0.  Thus  the  orginal  equation 
is  equal  to  these  4  separate  simple  equations,  and  the  roots  are 
1,  -2,  4,  5. 

Thus  a  biquadratic  equation  has  4  roots.  And  so  we 
might  go  on. 

The  general  rule  for  solving  an  equation  of  a  degree 


ALGEBRAIC  EQUATIONS  209 

beyond  the  first  is,  then,  to  bring  all  the  terms  to  the  left- 
hand  side,  to  reduce  the  resulting  expression  to  a  series  of 
linear  factors,  to  equate  each  of  these  to  0,  and  then  to  solve 
them  as  simple  equations. 

It  is  therefore  clear  that  the  roots  of  an  equation  of  any 
degree  may  be  written  down  at  once,  provided  we  can  resolve 
into  linear  factors  the  expression  which  results  from  bringing 
all  the  terms  of  the  equation  to  the  left-hand  side.  Generally 
speaking,  the  trouble  is  to  find  the  factors,  and  it  is  often 
necessary  to  resort  to  indirect  methods. 

The  Need  for  Verifying  Roots 

When  solving  equations,  we  frequently  adopt  the  device 
of  multiplying  or  dividing  both  sides  by  some  quantity, 
and  sometimes  we  square,  or  take  the  square  root  of,  each 
of  the  two  sides.  Is  this  always  allowable? 

Consider  an  equation  of  the  simplest  form,  one  having  only 
one  solution,  say  x  —  3  =  2.  Since  x  —  3  —  2,  x  —  5  =0, 
and  .-.  x  =  5.  Let  us  multiply  both  sides  of  the  original 
equation  by,  say,  (x  —  6).  Thus 

(x  -  3)  (X  -  6)  =  2(*  -  6), 
/.  x2  —  9x  -f  18  =  2.v  —  12, 
/.   x2  -  llx  H-  30  -  0, 
/.   (x  -  5)  (x  -  6)  =  0, 
.".   the  values  of  x  are  5  and  6. 

Thus  by  introducing  the  factor  (x  —  6)  we  have  trans- 
formed the  equation  into  another  completely  different.  The 
new  root  6  does  not  satisfy  the  original  equation. — Evidently 
when  we  have  solved  an  equation  we  must  see  if  the  roots 
really  satisfy  the  equation. 

Another  example:  consider  the  very  simple  equation 
x  =  3.  Square  both  sides, 

*2  =  9, 

/.  *2  _  9  =  0, 
/.  (x  +  3)  (x  -  3)  =  0. 

Hence  there  are  2  roots,  +3  and  —3,  as  compared  with 

(B291)  15 


210  CRAFTSMANSHIP  IN  MATHEMATICS 

only  one  root  (+3)  in   the  original   equation.      Thus  the 
squaring  has  introduced  an  extraneous  root. 
Another  example: 

3*  —  V*2^~2~4  =  16. 

/.  3*  -  16  =  V^-24 
:.  9*a  -  96*  +  256  =  x2  -  24 
/.  8*2  -  96*  -f  280  =  0, 
.'.  x2  -  I2x  +  35  =  0, 
...  (x  -  7)  (*  -  6)  =  0, 

/.  x  =  7  or  6. 

But  on  examination  we  find  that  only  7  satisfies  the  original 
equation;  5  does  not.  Hence  there  is  only  one  root.  We 
seem  to  have  solved  the  equation  in  the  usuial  way:  have  we 
done  anything  wrong?  Let  us  see  if  by  working  our  way  back- 
wards we  can  discover  any  sort  of  mistake. 

(x  -  5)  (x  -  7)  =  0, 

/.   x2  —  I2x  +  35  =  0. 

Multiplying  by  8,  8*2  —  96*  -f  280  =  0. 

Adding  x2  —  24  to  each  side,  9*2  —  96*  +  256  —  *2  -  24. 

Extracting  the  square  root  of  each  side,  3*  —  16  =  +  v/*2  —  24. 
.'.   3*  T   v^-l4  =  16. 

The  steps  are  exactly  the  same  until  we  come  to  the  last  but 
one;  then  we  had  to  prefix  the  double  sign.  Hence  at  the 
second  step  in  our  forward  process,  we  really  introduced  a 
new  and  extraneous  root,  since  the  square  of  +  V#2  —  24 
is  also  the  square  of  — V#2  — 24!  Thus  from  that  step  on- 
wards, the  equations  ceased  to  represent  the  original  equation. 
When  we  multiply  or  divide  by  ordinary  arithmetical 
numbers,  no  difficulty  will  arise.  When  we  multiply  or 
divide  by  an  algebraic  expression,  we  sometimes  run  a  risk. 
What  is  wrong  with  the  following,  for  instance? 

Suppose  *  =  y. 

Then  x2  =  ocy, 

.'.  x2  —  y2  —  xy  —  y*9 
.'•  (x  +  y)  (x  -  y)  =  *(*  -  y), 
:.  x  +  y  =  *, 
/.  x  -f  x  =  x, 

:.  2=1,  which  is  absurd. 


ALGEBRAIC  EQUATIONS  211 

We  have  divided  both  sides  in  the  fourth  line  by  (x  —  y), 
i.e.  by  (x  —  x),  i.e.  0.  This  is  quite  illegitimate,  and  it  in- 
evitably leads  to  an  absurdity.  Can  you  see  now  why  our 
first  example  went  wrong?  We  had,  really,  (x  —  5)  =  0; 
and  then  (x  •—  5)  (x  -—  6)  =  Q(x  —  6),  though  we  did  not 
show  it  this  way. 
Another  example: 

2        x   ,  1          A 

-f  2  4-   -----  =0. 


- 
x2  —  1  x  —  1 

Multiply  by    *2  -  1,  the  L.C.M., 

x*  ~  3x  -f  2(x*  -  1)  +  *  -h  1  =  0, 

/.   3*8  -  2x  -  1  -  0, 

/.  (3*  4-  1)  (x  -  1)  -  0, 

.*.  x  =  —  J  and  1. 

But  by  testing  we  find  that  1  is  not  a  value  of  the  original 
equation  and  must  therefore  be  rejected.  Multiplying  by 
(x2  —  i)  led  to  this  trouble.  Here  is  a  more  correct  way  of 
solving: 


»  -  3* 


-2  =  0, 
-^  +  2  =  0, 

j      o   rv 

.".  x  =  —  5,  the  only  root. 


The  former  method  is  quite  acceptable,  provided  the  roots 
found  are  checked,  and  that  one  is  rejected  if  found  un- 
acceptable. 

Strictly    speaking,     either  ...    or    are    "  disjunctive " 
they  therefore  suggest   alternatives.      But  sometimes  they 


212  CRAFTSMANSHIP  IN  MATHEMATICS 

are  equivalent  to  both  .  .  .  and]  or,  alike  .  .  .  and\  and  it 
is  this  exceptional  use  which  more  correctly  represents  the 
algebraic  argument.  But  the  use  of  either  ...  or  in  con- 
nexion with  equations  is  best  avoided.  When  a  boy  says 
"  either  6  or  5  "  he  naturally  thinks  that  if  one  is  accepted 
the  other  is  necessarily  rejected. 


The  Theory  of  Quadratics 

The  work  on  equations  should  be  closely  associated  with 
the  work  on  graphs.  The  graph  helps  to  elucidate  all  sorts 
of  difficulties.  See,  for  instance,  fig.  46,  p.  161,  in  connexion 
with  the  "  either  ...  or  "  argument. 

The  elementary  theory  of  quadratics,  as  far  as  it  is 
necessary  for  a  Fifth  Form,  seldom  gives  trouble.  The  more 
elementary  facts  should  be  known  thoroughly  and  should 
be  consistently  used  for  checking  and  other  purposes.  But 
do  not  forget  that  the  quadratic  function  is  of  far  greater 
importance  than  the  quadratic  equation  (see  the  chapter  on 

Graps)- 


.  -  ,      , 

The  formula  -  ==  ---    may  be  used  as   other 
2a 

formulae  are  used,  but  it  should  not  be  used  as  the  stock 
method  of  solving  quadratics;  boys  are  apt  to  forget  its 
significance  if  used  in  that  way.  They  should  clearly  realize 
that  the  formula  represents  the  roots  of  the  equation  ax2  +  bx 
+  c  —  0,  and  that  these  roots  are  real  and  different,  real 
and  equal,  or  unreal  and  different,  according  as  the  dis- 
criminant b2  —  4.ac  is  +,  0,  or  —  . 

The  pupils  should  frequently  make  use  of  the  further 
facts  that  if  xl  and  x2  arc  the  roots  of  the  equation  ax2  +  bx 

+  c  =  0,  then  x1  +  x2  =  —  ,  and  x^  =  -.     The  method 


a  a 


r        i      -         u        r          —  *  ±  Vb2  —  4ac    «      -j   , 
or  evaluating  these  from    -  —  -  should  be  re- 

membered. a 


ALGEBRAIC  EQUATIONS  213 

Equations  Solved  like  Quadratics 

Group  the  common  types  together.  They  are  usually 
easy,  though  attention  must  be  paid  to  all  the  roots  involved; 
the  boys  are  apt  to  overlook  some  of  them.  We  append 
examples  of  the  main  type,  and  add  the  sort  of  hint  that 
ought  to  suffice  to  enable  the  boys  to  set  to  work. 

1.  xl 


Write  *2  =  y,  and  solve  for  y\   then  from  the  2  values  of  y 
obtain  the  4  for  x. 

2.  (x2  +  2)2  -  29(*2  -J-  2)    |    198  -  0. 

Write  #2  +  2  =  y  and  solve  for  y. 

3.  2xz  -  4x  -f  3  Vx2  ~2x  -\   6  -  15  =  0. 

/.   2(x*  -  2*  -f  6)  +  3V*2  ^-"2*  +7>  -  27  =  0. 
Write  Vx2  —  2x  4-  6  =  y  and  solve  for  y. 

4.  (x  -l)(x-  3)  (x  -  5)  (x  -  7)  =  9, 

.'.  (x  —!)(*—  7)  (x  -  3)  (x  -  5)  -  9  -  0, 
/.  (x*  -  8*  +  7)  (^c2  -  8^  +  15)  -  9  =  0. 
Write  x2  —  8x  =  y,  and  solve  for  y. 

x  -f  4       A:  —  4       9  +  ^:       9  —  A: 


5. 


-  4       i  +  4       9  —"i       9  + 
16*  36* 


—  16       81  —  x2 
4 9 


6.  X*+  -  -  11  -0, 

/**2 


(obviously  *  —  0  is  a  root), 
(obviously  2  more  roots). 


/.  **  -  II*2  -f  18  =  0. 

7.  xz  +  1  =  0.         (Factorize.) 

8.  x6  —  7x*  —  8  =r  0.         (Factorize.) 

9.  7x*  -  I3xz  +  Zx  +  3  =  0.         (Factorize:  (*  -  1)  evidently 

a  factor.) 

Boys  soon  see  through  all  these  types  and  solve  examples 
fairly  readily. 


214  CRAFTSMANSHIP   IN  MATHEMATICS 

Simultaneous  Equations 

Do  not  spend  much  time  over  these,  unless  you  are  un- 
lucky enough  to  have  to  prepare  for  an  unintelligent  exam- 
ination in  which  far-fetched  examples  are  given.  Let  a 
result  of  each  main  type  be  graphed  (see  the  Chapter  on 
Graphs)  and  all  the  roots  be  pictorially  explained. 

Teach  the  pupils  to  pair  off  the  roots  correctly. 

We  append  an  example  of  the  commoner  types. 

1.  x  -\  y  =  1 

XV  —  14  Show  that  once  we  know  the  value  of  x  -\-  y 

and  x  —  y,  we  may  obtain  the  separate 

2.  x2  -f  y2  —-  1^  values  of  x  and  y  by  mere  addition  and 

xy  —  6  subtraction;  and  that  x  -\-  y  can  always  be 

obtained  from  x2  -f  2xy  ~|-  y2,  and  x  —  y 

3.  X  -  y  =  2  from  x*  _  2xy  +  yZf 


4.  x3  -!  y1  =  152 

X   \-  y    -  S  By  division  we  obtain  quotients  which 

enable  us  to  proceed  as  in  examples 

5.  *3-j,3  =  98  Ito3. 

x2  -f  xy  f  y2  =  49 

Ity  division,  we  obtain  x2  ~  xy  +  ya, 

6.  ^4  +  ^j;2  -\-y*  --  133        which,    subtracted    from   x2  -f  ^ 

Jt:2  4-  ^ry  +  jy2  ^  19  -I-  y2,   gives   us   ;ry.     Then   as   in 

examples  1  to  5. 

7.  3x*  +  4xy  +  5);2  ^=31     From  the  second  express  y  in  terms 

X  ~\-  2y  —  5  of  x,  and  substitute  in  the  first. 

Expressions  are  homogeneous.     Con- 
vert   into    fractions,    simplify,   and 

8.  X2  +  3xy  —y2  =  9        factorize.     Thus  we  have: 
2x2  -  2xy  -f  3y2  =  7  ll*a  -  39^  +  34>'2  =  0, 

/.   (11*-  lly)  (x  -  2y)  =  0, 
.'.  x  =  ^{y  and  2y\  &c. 


All  these  types  are  easily  taught  and  remembered.  It  is 
waste  of  time  for  boys  to  be  given  the  far-fetched  and 
exceptional  types  worked  out  (often  elegantly  it  is  true) 
in  the  textbooks.  School  life  is  not  long  enough. 


ALGEBRAIC  EQUATIONS  215 

Problems  producing  Equations 

These  have  been  given  a  place  greatly  beyond  their  value, 
and  important  mathematical  principles  are  often  treated 
rather  superficially  in  order  that  more  time  may  be  devoted 
to  "  problems  ".  Unfortunately,  however,  problems  have 
become  entrenched  in  all  mathematical  examinations,  and 
there  is  nothing  for  it  but  to  teach  boys  how  to  solve  them. 
And,  after  all,  problems  do  test  boys'  knowledge  of  certain 
principles,  and  a  correct  solution  is  always  a  source  of  satis- 
faction. 

The  veriest  tyro  of  a  teacher  can  write  out  on  the  black- 
board the  solution  of  a  problem  which  the  boys  themselves 
have  been  unable  to  solve.  But  what  do  the  boys  gain  from 
that?  The  mere  setting  out  of  a  solution  deductively,  after 
the  manner  of  a  proposition  in  Euclid,  gives  the  boys  no 
inner  light  at  all.  The  boys  want  to  be  initiated  into  a  plan 
of  effective  attack,  to  be  taught  how  to  analyse  and  how  to 
utilize  the  data  of  a  problem,  to  be  told  exactly  how  the 
teacher  himself  discovered  the  solution. 

Be  it  remembered  that  a  solution  most  suitable  for  a 
class  of  boys  is  by  no  means  necessarily  the  "  neat  "  solution 
so  dear  to  the  heart  of  a  mathematician. 

The  main  difficulty  felt  by  boys  in  solving  most  algebra 
problems  is  the  translation  of  the  words  of  the  problem 
into  suitable  equating  formulae.  Much  practice  is  necessary 
if  facility  in  this  translation  is  to  be  gained.  Once  expressed 
in  algebraic  form,  the  equation  is  generally  easy  of  solution, 

The  boy  knows  that  an  equation  consists  of  two  parts 
connected  together  by  the  sign  =.  The  first  thing  to  search 
for  in  a  given  problem  is  therefore  the  word  "  equal  ",  or 
some  words  which  imply  "  equal  ",  or  such  words  as  "  greater 
than  "  or  "  less  than  ".  If  the  problem  concerns  money 
matters,  the  boy  may  be  able  to  dig  out  of  his  own  know- 
ledge some  relation  of  equality,  e.g. 

Cost  price  +  profit  =  selling  price; 


216  CRAFTSMANSHIP  IN  MATHEMATICS 

or,  if  he  is  dealing  with  racing  problems,  he  may  be  able  to 
utilize  the  already  familiar  relation: 

distance  =  speed  X  time; 

or,  if  he  is  dealing  with  a  clock  sum,  he  may  be  able  to  split 
up  a  component  angle  in  two  different  ways,  and  so  equate 

a  +  P  —  y  ~f  8  (or  some  modification  of  this). 
We  append  a  few  problems,  with  teaching  hints. 

1.  If  4   be  added  to  a  certain  number,   and  the  sum  be 
multiplied  by  5,  the  product  will  be  equal  to  the  number  added 
to  32.    Find  the  number. 

The  question  tells  us, 

a  product  —  the  number  +  32 (i) 

Let  us  try  to  arrange  our  equation  accordingly. 

What  have  we  to  find?  A  number.  Then  let  x  represent 
the  number.  The  "  product  "  is  5  times  the  sum  of  x  and  4; 
how  shall  we  write  this  down?  5(#  +  4).  (i)  tells  us  that 
this  product  is  equal  to  x  +  32; 

/.   5(#  +  4)  =  x  +  32.  /.  x  =  3. 

2.  Find  a  number  such  that  if  it  be  multiplied  by  5,  and  2 
be  taken  from  the  product,  one-half  the  remainder  shall  exceed 
the  number  by  5. 

The  question  says 

half  a  remainder  exceeds  the  number  by  5, 
i.e.  half  a  remainder  =  the  number  -\-  5 (i) 

How  shall  we  represent  the  number?  by  x.  5  times  the 
number?  5x.  What  is  the  remainder  when  2  is  taken  from 
this  product?  6x  —  2.  What  is  half  this  remainder?  %(5x  —  2). 
Then  how  from  (i)  can  we  make  up  our  equation? 

-  2)  =  x  +  5.  .'.  x  =  4. 


ALGEBRAIC  EQUATIONS  217 

3.  A  man  spent  £10  of  his  money,  and  afterwards  one- 
quarter  of  the  remainder.    He  had  £30  left.      How  much  had 
he  at  first? 

The  word  left  suggests  the  -relation: 

(money  at  first)  —  (expenditure)  =  £30.     .     .     .     (i) 

Let  us  try  to  arrange  our  equation  accordingly. 

Let  x  represent  the  number  of  pounds  he  had  at  first.  Then 

x  —  expenditure  —  £30 (ii) 

What  is  the  expenditure? 

First  expenditure  =  £10;    /.  x  —  10  =  remainder. 
Second  expenditure  =  }  of  remainder  =  -J(*v  —  10), 
.'.   Total  expenditure  =  10  -f  \(x  —  10). 

Now  we  may  substitute  this  in  (ii): 

.'.  x  -  {10  +  \(x  -  10)}  -  30.       /.  X  =  £50. 

4.  A  man  buys  a  flock  of  sheep  at  £3  a  head,  and  turns 
them  into  a  field  to  graze  for  3  months,  for  which  he  is  charged 
45s.  a  score.     He  then  sells  them  at  £3,  10s.  a  head,  and  so 
makes  a  clear  profit  of  £77,  10s.    How  many  sheep  were  there 
in  the  flock? 

Clearly: 

(Money  laid  out)    f  (profit)  —  (Proceeds  of  sale).         .     (i) 

What  have  we  to  find?    The  number  of  sheep  bought. 

x 
Let  x  represent  no.  of  sheep  bought;   then  —  =  no.  of 

scores  of  sheep   bought. 

1.  Money  laid  out: 

(a)  Cost  of  x  sheep  at  £3  each  =  3x  pounds 

(b)  Cost  of  grazing  --  scores  of  sheep  at  £2 \  a  score  =  f  —  X  2 

/.  total  money  laid  out  =  3#  -f-  ( ~  X  2H .   .     .     (ii) 

2.  Proceeds  of  sale: 

Sale  of  x  sheep  at  £3  J  each  =  3J#  pounds.     ,     .     .    (iii) 


218  CRAFTSMANSHIP  IN  MATHEMATICS 

According  to  (i),  (ii)  +  £?7|  =  (iii), 

i.e.     3*  +    ~  X  2i    +  77J  -  3£*.          /.  X  -  200 


5.  A  boy  was  born  in  March.     On  the  18th  of  April  he 
was  5  times  as  many  days  old  as  the  month  of  March  was  on 
the  day  before  his  birth.    Find  his  date  of  birth.* 

This  examination  absurdity  is  simple  enough,  once  the 
wording  is  unravelled.  Note  that  if  a  boy  is  born,  say,  on 
4th  May,  he  is  20  days  old  on  24th  of  May.  It  is  a  case  of 
simple  subtraction. 

In  the  problem  we  have  to  deal  with  two  ages,  expressed 
in  days: 

(i)  the  age  of  the  boy  on  J8th  of  April, 

(ii)  the  age  of  March  on  the  day  before  the  boy  was  born 

The  former  =   5  times  the  latter.     Hence  we  can 

make  up  our  equation       ........  (i) 

(i)  The  age  of  the  boy  on  18th  of  April: 

Let  the  boy  be  born  on  the  #th  of  March. 

By  the  end  of  March  he  is  (31  —  x)  days  old. 

By  April  18th  he  is  (31  —  *  +  18)  days  old  .....     (ii) 

(ii)  The  age  of  March  on  the  day  before  the  boy  was  born. 

The  boy  was  born  on  the  #th  day  of  March. 

Hence  March  was  then  x  days  old. 

The  day  before  that,  March  was  (x  —  1)  days  old.  .     .     (iii) 

From  (i)  we  know  that  (ii)  is  5  times  (iii). 

i.e.     (31  -  x)  +  18  =  5(*  -  1).  /.  x  =  9. 

6.  The  3  hands  of  a  watch  are  all  pivoted  together  centrally. 
When  first  after  12.0  will  the  seconds  hand,  produced  back- 
wards, bisect  the  angle  between  the  other  2  hands? 

We  have  to  remember  that  the  seconds  hand  moves  60 

•  The  problem  is  not  well  worded.  For  instance,  March  is  not,  strictly,  nine 
days  old  until  midnight  on  March  Qth.  We  have  assumed  that  the  boy  was  born 
at  midnight,  and  we  have  reckoned  ages  from  midnight. 


ALGEBRAIC  EQUATIONS 


219 


times  as  fast  as  the  minute  hand,  and  the  minute  hand  12 
times  as  fast  as  the  hour  hand.  Thus  the  relative  speeds  are 
720  :  12  :  1. 

At   noon   (N)   all   the  N 

hands  are  together.  The 
watch  circumference  is 
divided  into  60  equal  arcs, 
and  we  may  measure  the 
angles  in  terms  of  these 
arcs.  Let  the  seconds 
hand  move  round  to  its 
position  S*  in  x  seconds; 
i.e.  arc  NS  =  x.  Since 
the  minute  hand  also 
moves  round  to  its  posi- 
tion M  in  x  seconds,  the 
x 
60' 

And  since  the  hour  hand  also  moves  round  to  its  position 

v 

H,  in  x  seconds,  the  arc  NH  measures  -— . 

Now  the  seconds  hand  OS  produced  backwards,  making 
OS',  bisects  the  angle  HOM;  i.e.  the  arc  HS'  =  the  arc  S'M. 

We  ought  therefore  to  be  able  to  make  up  an  equation 
by  means  of  the  pieces  of  arc  between  N  and  M,  e.g. 

NM  =  NS'  4-  S'M (i) 

We  know  that        NM  =  %-  , 


arc  NM  measures 


Fig.  61 


-  NH)  = 


that        NS'  =  x  -  30, 
and  that        S'M  =  JHM  - 

.'.  from  (i)  we  have        ~  =  (x  —  30)  +  |f  ~  —  -~\, 

:.  x  =  30^^  (sees,  after  12.0). 
•  The  angles  in  the  figure  are  necessarily  much  exaggerated. 


220  CRAFTSMANSHIP  IN  MATHEMATICS 

7.  54  minutes  ago,  it  was  5  times  as  many  minutes  past  5 
as  it  is  now  minutes  to  7.  What  is  the  time  now? 

Most  watch  and  clock  problems  can  be  solved  on  the  basis 
of  the  principle  illustrated  in  the  last  example,  and  one 
careful  analysis,  to  exemplify  the  method,  is  usually  enough 
to  enable  the  boys  to  attack  successfully  most  of  the  problems 
given  in  a  textbook.  But  this  problem,  another  absurdity 
from  an  examination  paper,  does  not  fit  into  any  general 
scheme.  Though  easy,  its  translation  into  an  equation  may 
at  first  puzzle  most  average  pupils. 

The  basis  for  equalizing  quantities  is  pretty  obvious  at 
the  outset: 

(54  min.  ago,  no.  of  min.  past  5)  =  5  (no.  of  min.  to  7  now).     .     (i) 

The  question  to  be  answered  is,  what  is  the  time  nowl 
The  problem  mentions  5.0  and  7.0,  and  refers  to  the  time 
now  as  a  number  of  minutes  to  seven. 

Hence,  let  the  time  now  be  x  minutes  to  7.       .     .     .     (ii) 

We  also  require  to  know  what  the  time  was  54  minutes 
ago\  this  must  have  been  (x  +  54)  minutes  to  7. 

But  we  have  to  express  this  time  in  terms  of  minutes 
past  5.0.  Now  5.0  is  120  minutes  before  7.0. 

Hence,  54  minutes  ago  the  number  of  minutes  past  5.0 

was  120  —  (x  -f  54) (iii) 

From  (i),  (iii)  is  5  times  (ii), 

i.e.  120  -  (x  -|-  54)  =  5*.  /.  x  =  11. 

All  answers  to  equations  should  be  checked;  checking  in  a 
case  like  this  is  particularly  necessary. 

Since  x=  11,  the  time  now  is  11  minutes  to  7,  or  6.49. 
The  time  54  minutes  ago  was  5.55,  or  the  number  of  minutes 
(55)  then  past  5.0  is  5  times  the  number  of  minutes  (11)  now 
to  7.0. 


ALGEBRAIC  EQUATIONS  221 

8.  Three  friends  going  on  a  railway  journey  take  with 
them  luggage  amounting  in  all  to  6  cwt.  Each  has  more  than 
can  be  carried  free,  and  the  excess  charged  them  is  2s.  6d.,  7s., 
and  10s.,  respectively.  Had  the  whole  belonged  to  one  person, 
he  would  have  had  to  pay  34s.  6d.  excess.  How  much  luggage 
is  each  passenger  allowed  to  carry  free,  what  is  the  excess 
charge  per  Ib.,  and  what  is  the  weight  carried  by  each  of  the 
three  friends? 

Consider  first  a  simple  case.  If  I  am  allowed  to  take 
with  me,  say,  100  Ib.  free,  and  have  to  pay,  say,  \d.  on  every 
Ib.  exceeding  100,  then  if  I  take  with  me  a  total  of,  say,  150  Ib. 
the  excess  I  have  to  pay  is  \d.  X  (150—  100). 

Thus  a  possible  form  of  equation  seems  to  be: 

(excess  charge  per  Ib.)    X  (no.  of  Ib.  excess)  =  (total  charge  for  excess), 

and  as  there  are  two  separate  though  similar  statements 
concerning  excess,  we  ought  to  be  able  to  formulate  two 
equations,  say  in  x  and  y. 

What  have  I  to  find  out?  (1)  Ib.  per  person  carried  free, 
and  (2)  excess  charge  per  Ib.  Hence: 

Let  each  passenger 

(1)  Carry  x  Ib.  free. 

(2)  Pay  y  pence  on  each  Ib.  excess. 

Assume  that  one  of  the  friends  takes  3  tickets  and  shows 
them  to  the  porter,  who  on  weighing  the  luggage  and  finding  it 
to  be  672  Ib.,  deducts  3x  from  the  672,  and  charges  y  pence  per 
Ib.  on  the  difference,  viz.  (672  —  3#).  Since  the  sum  actually 
paid  for  excess  =  2s.  6d.  +  7*.  +  10s.  =  19s.  6d.  =  234^.,  , 

/.   (672  -  3x)y  =  234 (i) 

But  if  all  672  Ib.  had  belonged  to  one  person,  he  would 
have  taken  only  1  ticket,  and  the  porter  would  have  charged 
y  pence  on  each  of  (672  —  x)  Ib.  Since  the  sum  actually  paid 
in  this  case  for  excess  =  34s.  Qd.  =  414J., 

.'.  (672  -  x)y  =  414 (ii) 


222  CRAFTSMANSHIP  IN  MATHEMATICS 

Dividing  (i)  by  (ii)  we  have 

672  -  3x  _  13 

672- *        23'  /.  #=  120 

=  no.  of  Ib.  allowed  free. 
Hence  from  (i)  or  (ii), 

y  —  \d.  =  charge  per  Ib.  excess. 
Weight  in  Ibs.  carried  by  each  person 

=   120  +  ~,   120  +  8*     120  +  1|?,  respectively. 
Ill 

This  problem  is  worth  giving  a  class  a  second  time,  say 
three  months  after  the  first.  Boys  seem  to  find  a  first  analysis 
a  little  difficult. 

9.  An  express  train  and  an  ordinary  train  travel  from 
London  to  Carlisle,  a  distance  of  300  miles.  The  ordinary 
train  loses  as  much  time  in  stoppages  as  it  takes  to  travel  25 
miles  without  stopping.  The  express  train  loses  only  three- 
tenths  as  much  time  in  stoppages  as  the  ordinary  train ,  and  it 
also  travels  20  miles  an  hour  quicker.  The  total  times  of  the 
two  trains  on  the  journey  are  in  the  ratio  26  :  15.  Find  the 
rate  of  each  train. 

We  have  to  think  of  distance,  speed,  and  time,  and  their 
relation  d  =  st. 

The  ratio  of  the  total  times  taken  by  the  two  trains  is 

given,  viz.  26:15 (i) 

Hence  if  we  can  express  these  total  times  in  some  other  way 
we  can  formulate  our  equation. 

Distance:  300  miles. 

Speeds'.  Let  ordinary  train  travel  x  miles  an  hour. 

Then  express  train  travels  x  -f  20  miles  an  hour. 

Qf)f) 

Times:   Time  taken  by  ordinary  train  = f-  stoppages  (in 

x 

hours) (ii) 

Time  taken  by  express  train  = — -  +  stoppages  (in 

X  ~j~  20 

hours) .     .     (iii) 


ALGEBRAIC  EQUATIONS  223 

We   now  require  to  know  the   amount   of  time   lost  over 
stoppages. 

(1)  Ordinary  train:    time  lost  over  stoppages  is  equal  to 
that  taken  in  travelling  25  miles: 

Train  travels  x  miles  in  1  hour, 

=  1  mile  in  -  hour 
x 

25 

=  25   miles  in  —  hours 
X 

25 

.*.    -    hours  -  -  time  lost  over  stoppages. 
x 

(2)  Express  train:   time  lost  over  stoppages  =  -^  that  of 
ordinary  train, 

3     r 25  ,  15  , 

—  _   of  —  hours  =     —  hours. 
10        x  2* 

We  can  now  express  (ii)  and  (iii)  in  the  following  forms: 

Time  taken  by  ordinary  train  =  (  .  —  -|-  — )  hours.      .     (iv) 

\  x          x  ' 

rr-  t  t  .  /       300  .       15\    ,  ,     . 

Time  taken  by  express  tram  =  I -f-       J  hours.        (v) 

\x  -f  20       2x' 

Hence  we  have  from  (i),  (iv),  (v), 
300       25 

~x          x  26 


15        15 


*  +  20       2x 

.'.  X  =  30  (miles  an  hour). 

If  in  "  racing  "  and  analogous  problems  the  relation  d  —  st 
is  kept  in  view,  the  necessary  analysis  is  usually  quite  simple. 

10.  What  is  the  price  of  sheep  per  100  when  10  more  in 
£100  worth  lowers  the  price  by  £50  per  100? 

We  must  avoid  confusion  between  (1)  the  number  of 
sheep  for  £100  and  (2)  the  cost  of  100  sheep. 

We  can  find  the  number  of  sheep  costing  £100  if  we 
know  the  price  of  1,  and  we  can  find  the  price  of  1  if  we  know 
the  price  of  100. 


224  CRAFTSMANSHIP  IN  MATHEMATICS 

A  possible  equation  seems  to  be: 

(First  no.  of  sheep  for  £100) 

=  (second  no.  of  sheep  for  £100)  —  10.     .     .     (i) 

What  have  we  to  find?    The  price  of  100  sheep. 

(1)  First  price  of  100  sheep.    Call  this  £#. 

:.   1  sheep  costs  ^  , 

/.   number  obtainable  for  £100  =  £122  =  10'000.      .     .     (ii) 

&L  x 

100 

(2)  Second  price  of  100  sheep.    This  =  £(#  —  50). 
/.    1  sheep  costs  £(*  "  50). 

.-.  number  obtainable  for  £100  = 


(#  —50)       x  —  50 


100 


(i)  shows  us  how  (ii)  and  (iii)  are  related,  and  then  we  may 
make  up  our  equation. 

-  10,  ...  *  =  250. 


x  x  —  50 

i.e.  the  price  of  100  sheep  is  £250,  or  £2,  10*.  each,  or  £100 
worth  —  40  sheep. 

(If  50  for  £100,  each  costs  £2,  or  cost  of  100  =  £200, 
i.e.  £50  less  than  before.) 

I  have  found  that  even  Sixth  Form  boys  are  sometimes 
baffled  by  the  analysis  of  this  little  problem. 

Problems  which  are  at  all  unusual  in  form  are  always 
worth  repeating  after  an  interval. 

Books  to  consult  (on  the  general  technique  of  teaching  algebra): 

1.  The  Teaching  of  Algebra,  Nunn. 

2.  Elements  of  Algebra,  2  vols.,  Carson  and  Smith. 

3.  A  New  Algebra,  Barnard  and  Child. 

4.  Algebra,  Godfrey  and  Siddons. 

6.  A  General  Textbook  of  Elementary  Algebra,  Chapman. 
6.  Elements  of  Algebra,  De  Morgan.     (A  valuable  old  book.     So 
are  De  Morgan's  other  books,  especially  his  Arithmetic.) 


ELEMENTARY  GEOMETRY  225 

CHAPTER  XX 

Elementary  Geometry 

Early  Work 

Some  of  the  younger  generation  of  teachers  have  never 
read  Euclid,  and  seem  to  be  totally  unacquainted  with  the 
rigorous  logic  of  the  old  type  of  geometry  lesson.  Not  a 
few  of  the  old  generation  regret  the  disappearance  of  Euclid, 
urging  that  the  advantages  of  the  newer  work  are  outbalanced 
by  the  loss  of  the  advantages  of  the  older. 

The  real  distinction  between  the  older  and  the  newer 
work  is,  however,  sometimes  forgotten.  Essentially,  Euclid 
wrote  a  book  on  logic,  using  elementary  geometry  as  his 
raw  material.  The  amount  of  actual  geometry,  qua  geometry, 
which  he  taught  was,  relatively  speaking,  trifling.  Boys  in 
existing  technical  schools  do  ten  times  as  much  geometry 
as  is  found  in  Euclid.  But  as  an  exposition  of  deductive 
reasoning  from  an  accepted  set  of  first  principles,  Euclid  has 
never  been  equalled. 

Until  forty  years  ago,  Euclid  was  universally  taught  in 
secondary  schools,  but  the  collective  opinion  of  experts  had 
gradually  hardened  against  it,  partly  because  the  average 
boy  found  it  difficult,  partly  because  some  of  its  propositions 
were  too  subtle  for  schoolboys,  partly  because  its  foundations 
were  far  from  being  unassailable,  and  partly  because  the 
actual  geometry  it  expounded  was  too  slight  to  be  of  much 
practical  service. 

But  the  geometry  that  was  substituted  for  Euclid — the 
geometry  now  exemplified  in  all  the  ordinary  school  text- 
books— is  still  Euclidean  geometry,  i.e.  it  is  a  geometry  based, 
in  the  main,  on  the  same  foundations  as  Euclid.  These 
foundations  consist  of  a  number  of  quite  arbitrarily  chosen 
axioms.  Other  sets  of  axioms  might  be  substituted  for  them, 
and  then  we  should  get  an  entirely  new  system  of  geometry 

(B291)  16 


226  CRAFTSMANSHIP  IN  MATHEMATICS 

of  a  non-Euclidean  character.  Reference  to  such  geometry 
will  be  made  in  a  future  chapter. 

In  practice,  the  difference  between  Euclid  and  the  geo- 
metry now  taught  is  in  the  choice  of  working  tools.  In  Euclid, 
the  proof  of  every  proposition  was  ultimately  traceable  to 
the  axioms,  and  every  schoolboy  had  to  substantiate  every 
statement  he  made  by  referring  it  to  something  already  proved, 
and  this  in  its  turn  to  something  that  had  gone  before,  and 
so  back  to  the  axioms.  In  those  days  the  axioms  were  really 
the  working  tools.  But  those  axioms  were  so  subtle  that 
the  boys'  confidence  in  them  was  entirely  misplaced.  Now- 
adays, the  working  tools  consist  of  a  small  number  of  funda- 
mental propositions.  By  means  of  carefully  selected  forms 
of  practical  work,  the  truth  of  these  propositions  is  shown 
to  beginners  to  be  highly  probable,  but  the  formal  proofs 
of  such  propositions  are  not  considered  until  the  boys  reach 
the  Sixth  Form.  Examination  authorities  no  longer  call  for 
the  formal  proofs  at  the  School  Certificate  stage. 

These  working  tools  once  thoroughly  mastered,  beginners 
plunge  into  the  heart  of  the  subject  and  make  rapid  headway. 
In  the  old  Euclidean  days  a  year  or  more  was  spent  on  these 
propositions  and  a  few  others,  and  at  the  end  of  that  time 
the  average  boy  had  but  very  vague  notions  about  them, 
though  the  mathematically-minded  boy  certainly  did  seem 
to  appreciate  the  rigour  of  the  reasoning  presented  to  him. 

I  find  it  a  little  difficult  to  describe  the  methods  of  the 
pre-eminently  successful  teacher  of  geometry.  The  methods 
are  not  the  reflection  of  any  particular  book  but  of  the  man 
himself.  By  the  gifted  teacher  who  happens  to  be  a  sound 
mathematician,  a  new  principle  is  often  illuminated  by  so 
many  side-lights  that  even  the  dullard  can  hardly  fail  to  see 
and  understand.  Successful  teachers  of  geometry  seem  to 
be  those  who  have  given  special  attention  to  the  foundations 
of  the  subject,  who  possess  exceptional  ingenuity  in  making 
things  clear,  and  who  at  an  early  stage  make  use  of  symmetry 
and  of  proportion  and  similarity. 

Those  teachers  who  are  not  successful  are  often  those 


ELEMENTARY  GEOMETRY  227 

who  confine  their  work  to  the  limits  of  the  ordinary  text- 
book written  for  the  use  of  boys;  who  fail  to  survey  the  whole 
geometrical  field;  who  are  still  unacquainted  with,  or  at  all 
events  do  not  teach,  the  great  unifying  principles  of  geometry 
— duality,  continuity,  symmetry,  and  so  forth.  The  little 
textbooks  are  all  right  for  the  boys,  but  the  teaching  of  geo- 
metry connotes  something  outside  and  beyond  such  text- 
books, especially  the  principles  underlying  the  grouping  and 
regrouping  of  the  thousand  and  one  facts  that  the  beginner 
necessarily  learns  as  facts  more  or  less  isolated.  The  ac- 
cumulated facts  can  be  given  many  different  settings,  each 
setting  forming  a  perfect  picture,  all  the  pictures  different 
yet  closely  related. 

Work  up  to  13  or  14 

As  I  have  said  in  another  place,5*  the  following  is  an 
expression  of  authoritative  opinion  as  to  the  nature  of  the 

work  which  it  is  most  advisable  to  do  with  young  boys: 
/ 

1.  The  main  thing  should  «be  to  give  the  boys  an  in- 
telligent (knowledge  of  the  elementary  facts  of  geometry  A 

2.  No  attempt  should  be  made  to  develop  the  suoject 
on  rigorously  deductive  lines,  from  first  principles,  though, 
right   from  the   first,  /precise   reasons   for  statement^  made 
should  be  demanded. 

3.  Young  boys  are  never  happy  and  are  often  suspicious 
if  they  feel  they  are  being  asked  to  prove  the  obvious,  but  they 
can  follow  a  fairly  long  chain  of  reasoning  if  the  facts  are 
clear. 

4.  All  subtleties  should  be  avoided,  and,  therefore,  proofs 
of  propositions  concerning  angles  at  a  point,  parallels,  and 
congruent  triangles  should  not  be   attempted,  such  proofs 
being  a  matter  for  later  treatment  in  the  upper  Forms. 

5.  These  main  working  tools  of  geometry,  angles  at  a 
point,  parallels,  and  congruent  triangles,  should  be  presented 

*  Lower  and  Middle  Form  Geometry,  Preface. 


228  CRAFTSMANSHIP  IN  MATHEMATICS 

,  gJ^ 

in  such  a  way  as  to  $nable  the  boys  to  understand  them 
clearly  and  to  use  and  apply  them  readily. 

6.  Young  boys   can   easily   understand   Pythagoras,   ele- 
mentary facts  about  areas,  and  the  main  properties  of  the 
circle  and  of  polygons;    and  these  facts  should  be  taught. 

7.  The  simple  commensurable  treatment  of  (i)  the  pro- 
portional division  of  lines,  and  (ii)  similar  triangles,  should 
be  included  in  the  work  to  be  done  at  the  age  of  12  to  14; 
young  boys  soon  become  expert  in  the  useful  practice  of 
writing  down  equated  ratios  from  similar  triangles. 

8.  By  about  the  age  of  13,  a  boy  ought  to  be  able  to  write 
out  a  simple  straightforward  proof  formally  and  to  attack 
easy  riders. 

9.  Throughout  the  course,  all  possible  use  should  be  made 
of  the   boys '( intuitions   and   of  their   knowledge   of  space- 
relations  in  practical  life,)  relations  in  three  dimensions  as 
well  as  in  two. 

10.  Responsible   teachers   should   always   express   them- 
selves in  exact  (geometrical  language,  and  should  make  pre- 
cision  and  accuracy  of  statement  an   essential)  part  of  the 
boys'  training. 

11.  The  boys  should  be  taught  how  to  formulate  their 
own  definitions,  and,  under  the  guidance  of  the  teacher,  to 
polish  up  these  definitions  as  accurately  as  their  knowledge 
at  that  stage  permits;   and  these  definitions  should  be  learnt. 
Definitions  should  never  be  provided  ready-made. 

12.  The  boys  should  be  taught  to  realize  exactly  what 
properties  are  implied  by  each  definition,  and  all  other  pro- 
perties must  be  regarded  as  derivative  properties  requiring 
proof. 

13.  The  boys  may  usefully  be  given  an  elementary  training 
in  the  principle  that  a  general  figure  necessarily  retains  its 
basic   properties    even    when    it    becomes    more    and    more 
particularized,  but  that,  as  the  figure  becomes  less  and  less 
general,  it   acquires   more   and   more   properties;    and   vice 
versa. 

14.  In  short,  clear  notions  of  the  all-important  principle 


ELEMENTARY  GEOMETRY  229 

of  continuity  should,  by  the  time  a  boy  is  about  13,  "  be  in 
his  very  bones  ". 

15.  A  young  boy's  natural  fondness  for  puzzles  of  all 
kinds   may   often   usefully   be   employed   for   furthering  his 
interest  in  geometry. 

16.  In  one  respect  we  have  drifted  too  far  away  from 
Euclid:    boys'  knowledge  of  geometry  is  too  often  vague, 
too  seldom  exact. 

Teachers  differ  in  opinion  about  the  degree  of  accuracy 
to  be  demanded  in  beginners'  geometrical  drawing.  Some 
training  in  the  careful  use  of  instruments  is  certainly  desirable, 
but  time  should  not  be  wasted  over  elaborate  drawings 
when  freehand  sketches  can  be  made  to  serve  adequately. 
In  Technical  schools,  accurate  drawing  with  instruments  is  an 
essential  part  of  much  of  the  pupils'  work.  Even  in  Secondary 
schools,  where  the  geometry  is  necessarily  given  an  academic 
bias,  a  preliminary  training  in  the  careful  use  of  instruments 
serves  a  useful  purpose,  but  there  is  no  point  in  making  Secon 
dary  school  boys  spend  time  over  elaborate  pattern  drawings 
and  designs.  Exercises  in  accurate  work  of  a  more  telling 
type  may  be  found  in  the  theorems  of  Brianchon,  Desargues, 
Pascal,  and  others.  As  theorems,  these  are,  of  course,  work 
for  the  Upper  Forms;  as  geometrical  constructions,  they 
are  useful  in  the  lower  Middle  Forms,  where  they  may  be 
learnt  as  useful  and  interesting  geometrical  facts. 

All  pupils  should  be  taught  the  wisdom  of  drawing  good 
figures  for  rider  solving  purposes. 

Boys  in  Technical  schools  often  have  a  better  all-round 
knowledge  of  geometry  than  those  in  Secondary  schools 
because  they  do  more  work  in  three  dimensions.  Solid 
geometry  of  a  simple  kind  may  with  great  advantage  be 
included  in  the  early  stages  of  any  geometry  course. 

A  short  course  on  simple  projection  at  about  the  age  of 
13  helps  later  geometry  enormously.  Boys  soon  pick  up  the 
main  principles,  and  the  work  helps  greatly  to  develop  their 
geometrical  imagination.  So  does  simple  work  with  the 
polyhedra,  work  which  always  appeals  to  boys. 


230  CRAFTSMANSHIP  IN  MATHEMATICS 

Let  logic  of  the  strictly  formal  kind  wait  until  foundations 
are  well  and  truly  laid.  The  increasing  difficulty  felt  by 
beginners  in  geometry  is  largely  an  affair  of  increasing  diffi- 
culty of  logic,  and  thus  we  now  recognize  that  parts  of  the 
third,  fourth,  and  sixth  books  of  Euclid  are  easier  than  parts 
of  the  first  book. 

Push  ahead.  Do  not  paddle  about  year  after  year  in  the 
little  geometrical  pond  where  examiners  fish  for  their  questions. 
Even  for  the  examination  day  such  paddling  most  certainly 
does  not  pay. 

It  is  convenient,  though  not  defensible,  to  preserve  the 
old  distinction  between  axioms,  postulates,  and  definitions. 
But  if  any  teacher  still  believes  that  axioms  should  be  con- 
sidered in  a  beginner's  course  of  geometry,  let  him  consult 
Mr.  Bertrand  Russell,  and  he  will  soon  be  disabused. 

Early  Lessons 

Here  are  a  few  early  lessons  in  geometry,  selected  at 
random  from  the  book  already  cited.  The  sections  have 
been  renumbered,  seriatim,  for  convenience  of  reference,  but 
actually  the  lessons  are  drawn  from  all  parts  of  the  book. 

LESSON   I 
Planes  and  Perpendiculars 

1.  Carpenters,  bricklayers,  blacksmiths,  and  plumbers,  all 
have  to  know  something  about  geometry.  Architects,  builders, 
surveyors,  and  engineers  have  to  know  a  great  deal  about  it. 
All  of  them  have  to  know  how  to  measure  things,  and  how 
to  make  things  perfectly  level,  perfectly  upright,  perfectly 
square,  perfectly  "  true  ";  and  much  more  besides. 

You  have  already  learnt  how  to  use  a  ruler  or  scale, 
marked  with  inches  and  parts  of  an  inch  on  the  one  edge, 
and  with  centimetres  and  parts  of  a  centimetre  on  the  other. 
Note  that  there  are  very  nearly,  but  not  exactly,  2J  centi- 
metres to  an  inch. 


ELEMENTARY  GEOMETRY 


231 


You  know  already  that,  when  two  lines  meet  at  a  point, 
they  form  an  angle.     Here  are  three  angles. 


blade 


Fig.  63 


The  middle  one  is  the  angle  you  know  best.  It  is  the  angle 
you  see  at  the  corner  of  an  ordinary  picture-frame,  or  of  a 
door,  or  of  a  window-frame,  or  of  a  table-top.  Such  an  angle 
is  called  a  square  angle,  or  right  angle. 

2.  You  have  probably  seen  a  carpenter  planing  a  piece 
of  wood,  perhaps  for  a  shelf.     He  begins 

by  planing  one  of  the  "  faces  "  of  the  wood, 
and,  as  soon  as  he  thinks  that  the  face  is  a 
true  plane,  he  tests  it.  To  do  this  he  uses 
a  try -square,  which  consists  of  a  steel 
blade  with  parallel  edges  perfectly  straight, 
fixed  at  right  angles  into  a  wooden  stock. 
(A  /ry-square  is  a  tatf-square.)  He  holds  the  stock  in  his 
hand,  and  to  the  planed  face  of  the  wood  he  applies  the 
outside  edge  of  the  blade,  "  trying  "  it  in  many  places  and  in 
different  directions,  along  and  across.  If  he  can  see  day- 
light anywhere  between  the  blade  and  the  wood,  he  knows 
that  the  planed  face  is  not  yet  a  true  plane,  and  that  he 
must  continue  his  planing.  When  it  is  true,  he  marks  it 
face- side. 

3.  Now  he  turns  up  the  wood  so  that 
an  edge  rests  on  the  bench,  and  he  planes 
the  edge  at  the  top.    Not  only  has  he  to 
make  this  face-edge  (as  it  is  called)  a  true 
plane  like  the  face-side,  but  he  has  to  make 
the  two  planes  at  right  angles  to  each  other, 

or,  as  we  usually  say,  perpendicular  to  each  other.  The 
carpenter  is  not  satisfied  until  the  inside  right  angle  of  his 
try-square  fits  exactly,  at  the  same  time,  the  face-side  and 


Fig.  64 


232  CRAFTSMANSHIP  IN  MATHEMATICS 

the  face-edge,  the  blade  fitting  against  the  one,  the  stock 
against  the  other,  the  test  being  made  at  several  places  along 
the  wood. 

You  might  use  a  big  try-square  to  see  if  a  flag-staff  or  a 
telegraph-post  is  perpendicular  to  the  ground.  If  it  did  not 
fit  exactly  in  the  angle  between  the  post  and  the'  ground, 
no  matter  where  tested  round  the  post,  you  would  know 
that  either  the  ground  is  not  level  or  the  post  is  not  upright. 
Two  planes,  or  two  lines,  or  a  line  and  a  plane,  are 
perpendicular  to  each  other  if  and  only  if  they  are 
at  right  angles  to  each  other.  A  perfectly  upright  post 
in  a  sloping  bank  is  not  perpendicular  to  the  bank  because 
it  does  not  make  right  angles  with  the  bank. 

4.  The  maker  of  a  try-square  guarantees  the  accuracy 
of  the  inside  angle,  but  not  of  the  outside  right  angle.  Thus 
you  may  use  it  for  testing  the  right  angles 
of  a  table-top,  of  a  door,  of  the  outside  of 
a  box.  It  is  not  advisable  to  use  it  for 
testing  the  inside  right  angles  of  a  box,  or 


i^g.  6S  of  a  drawer,  or  of  a  door-frame.  (Strictly 

speaking,  it  ought  not  to  have  been  used 
for  testing  the  right  angle  round  the  flag-staff  or  telegraph- 
post.)  For  testing  inside  right  angles,  we  use  an  architect's 
set-square,  a  flat  triangular  piece  of  wood  with  a  true  right 
angle.  You  will  be  given  two  of  these  to  work  with,  a  little 
later  on. 

5.  You  have  learnt  that,  when  you  want  to  find  out  if  a 
surface  is  a  true  plane,  you  must  test  it  with  an  accurately 
made  straight-edge  of  some  kind.  If,  on  the  other  hand, 
you  are  doubtful  about  the  accuracy  of  a  straight-edge  (an 
ordinary  ruler,  for  instance),  you  can  test  it  by  applying 
it  to  a  plane  known  to  be  true.  Thus,  a  true  straight-edge 
may  be  used  for  testing  a  plane,  and  a  true  plane  may  be 
used  for  testing  a  straight-edge.  One  must  be  true,  and  then 
it  may  be  used  for  testing  the  other. 

(When  numbered  statements  in  dark  type  are  followed  by  the 
letter  "  L  ",  the  statements  are  to  be  learnt,  perfectly.) 


ELEMENTARY  GEOMETRY  233 

6.  A  PLANE  SURFACE  (or  a  PLANE)  is  a  surface 
in   which   a   true   straight-edge   will    everywhere   fit 
exactly.    (L.) 

LESSON  II 

Horizontal,  Vertical,  and  Oblique  Lines  and  Planes 

7.  Borrow  a  spirit-level   from  the  Geography  Master, 
or  from  the  school  carpenter,  and  see  if  the  floor  of  your  room, 
the  top  of  the  table,  the  window  ledge,  and  the  mantelpiece 
are  horizontal  (perfectly  level).     (Your  master  will  explain 
how  the  spirit-level  is  made  and  used.)     If  a  plane  surface 
is  everywhere  horizontal,  the  surface  is  called  a  horizontal 
plane,  and  straight  lines  drawn  on  that  surface  are  horizontal 
lines.    The  surface  of  still  water  (in  a  basin,  for  instance)  is 
a  horizontal  plane,  and  floating  lead-pencils  may  be  regarded  as 
representing  horizontal  lines.     The  edge  of  a  book-shelf,  the 
edge  of  a  table- top,  the  joints  of  floorboards,  the  line  where  the 
floor  meets  a  wall,  are  other  examples  of  horizontal  lines. 

8.  You  have  probably  seen  a  bricklayer  use  a  plumb- 
line— a  cord  stretched  straight  by  a  hanging  leaden  weight. 
He  uses  it  to  see  if  the  walls  he  is  building  are  vertical 
(perfectly  upright).    Make  a  plumb-line  for  yourself,  and  see 
if  your  school  walls  are  vertical.     If  they  are  vertical  and  if 
they  are  plane,  their  surfaces  are  vertical  planes.     Cover 
the  plumb-line  with  chalk,  hold  it  close  to  the  wall  and  let 
it  come  to  rest,  then  pull  it  out  towards  you  a  little  way  and 
let  it  go  suddenly.     It  springs  back  and  leaves  a  straight 
chalk-line  on  the  wall.    This  straight  line  is  a  vertical  line. 
The    balusters    on    a    stair-case,    hanging    chains,    hanging 
ropes,  telegraph  poles,  the  lines  where  any  two  walls  of  a 
room  meet  each  other,  may  all  be  regarded  as  representing 
vertical  lines.     Rain-drops  fall  in  vertical  lines,  unless  there 
is  a  wind.     A  telegraph  post  fixed  in  horizontal  ground  is 
both  vertical  and  perpendicular;    if  fixed  in  a  sloping  bank, 
it  is  vertical  but  not  perpendicular.     Why? 

Vertical    lines    always    point    downwards,    towards    the 
centre  of  the  earth. 


234  CRAFTSMANSHIP  IN  MATHEMATICS 

Butterflies    alight    with    their    wings    in    vertical    planes, 
moths  with  their  wings  in  horizontal  planes. 

9.  Planes  and  straight   lines  which  are  neither  vertical 
nor  horizontal  are  called  oblique.    Oblique  means  slanting 
or  sloping. 

10.  On  the  vertical  surface  of  a  wall,  it  is  easy  enough  to 
draw   both   horizontal   and   vertical   and   oblique   lines,   but 
vertical  and  oblique  lines  cannot  be  drawn  on  a  horizontal 
sheet  of  paper  lying  on  the  table.    All  lines  on  a  horizontal 
plane  are  horizontal.    Yet  it  would  be  inconvenient  to  have 
to  draw  lines  on  a  sheet  of  paper  which  is  pinned  to  the 
wall,   though   sometimes   your   master   certainly   does   draw 
lines  on  a  vertical  blackboard.     It  has  been  decided,  just  as 
a   matter   of  convenience   when   drawing,   to   represent   the 
three   different   kinds   of  straight   lines   all   on   a   horizontal 
plane,  and  in  this  way:    horizontal  lines,  parallel  to  the  top 
and   bottom   edges   of  your   paper;    vertical   lines,   parallel 
to  the  left-  and  right-hand  edges  of  your  paper;    oblique 
lines,  lines  in  any  other  direction. 


Horizontal  lines  Vertical  lines  Oblique  lines 

But  remember  that,  as  long  as  your  paper  is  lying  on  the 
horizontal  table,  it  is  not  strictly  true  to  say  that  the  lines 
you  draw  on  it  are  anything  but  horizontal.  We  do  not 
obtain  a  true  picture  unless  we  hold  the  paper  in  a  vertical 
plane  (against  the  wall,  for  instance).  Then  the  vertical 
lines  may  be  made  to  appear  really  vertical. 

11.  HORIZONTAL  PLANES  are  planes  which  are 
perfectly  level.    (L.) 

12.  HORIZONTAL  LINES  are  straight  lines  in  a 
horizontal  plane.    (L.) 

13.  VERTICAL    PLANES    are   planes   which   are 
perfectly  upright.    (L.) 


ELEMENTARY  GEOMETRY 


235 


Fig.  66 


14.  VERTICAL    LINES    are    straight    lines    in    a 
vertical    plane    that    point   downwards    towards    the 
centre  of  the  earth.    (L.) 

15.  Planes  and   straight   lines   which   are    neither 
horizontal  nor  vertical  are  called  OBLIQUE.    (L.) 

LESSON   III 
Solids  and  Surfaces 

16.  Here  is  a  brick.    Measure  it.    It  is  9"  long,  41"  broad, 
3"  thick.     It  is  a  solid  body,  but,  in  geometry,  we  call  it  a 
solid  not  because  it  is  made  throughout  of  a 
particular   kind   of  hard  stuff  but   because   it 

occupies  a  certain  amount  of  space.     If  the 

brick  were  hollow  and  made  of  paper,  we  should 

still  call  it,  in  our  geometry  lessons,  a  solid. 

A  room  of  a  house  is  a  solid;  so  is  an  empty  box.     Both  have 

length,  breadth,  and  thickness.     But  we  do  not  usually 

speak  of  the  thickness  of  a  house  or  of  a  box.    We  say  that  a 

house  has  length,  breadth,  and  height,  and  a  box  length, 

breadth,  and  depth.    But  all  have  three  dimensions;  that  is, 

we  can  measure  them  from  front  to  back,  from  side  to  side, 

and  from  top  to  bottom.    (Both  the  word  dimension  and  the 

word  mensuration  are  derived  from 

the  same   Latin  word,  mensura,  a 

measure.) 

17.  Here  are  a  cube  (fig.  67,  i) 
and   a    square   prism   (ii).     You 
have  probably  seen   them   before, 
and      know      their 

names.  If  they 
were  made  of  paper, 
we  could  run  a 
knife  along  some  of 
their  edges  and  lay 
them  out  flat  like 

this:  Fig.  68 


Fig.  67 


236 


CRAFTSMANSHIP  IN  MATHEMATICS 


Plans  of  this  kind  are  called  the  nets  of  the  solids.  Later 
on  in  the  geometry  book,  you  will  find  instructions  how  to  cut 
out  nets  from  stiff  paper  and  how  to  fold  and  bind  them 
up  into  the  solids  they  represent. 

18.  The  surface  of  both  the  cube  and  the  prism  consists 
of  six  faces.    All  six  faces  of  the  cube  are  squares.    Only 
two  faces  of  the  prism  are  squares,  the  other  four  being 
oblongs.     Sometimes  we  speak  of  the  two  square  faces  of 
the  prism  as  ends  or  bases,  and  the  four  oblong  faces  as 
the  sides.    In  each  case,  all  the  faces  are,  of  course,  planes. 
Any  two  adjoining  planes  of  the  cube  or  of  the  prism  meet 
in  an  edge,  or,  as  we  sometimes  say  in  geometry,  the  two 
planes  intersect  in  a  straight  line. 

19.  Here  are  four  more  solids  which  you  have  probably 
seen  before;    a  square  pyramid,  a  cylinder,  a  cone,  and 
a  sphere. 


Fig.  69 


20.  In  shape,  the  square  pyramid  reminds  you  of  the 
famous  Egyptian  pyramid.    Its  surface  consists  of  five  plane 
faces,   namely,   one  square   base,   and   four  triangular  faces 
meeting  in  a  point  called  the  vertex.    The  vertex  is  exactly 
over  the  centre  of  the  base  *  (fig.  69,  i). 

21.  The  cylinder  reminds  you  of  a  garden  roller,  of  a 
jam-jar,  or  of  part  of  a  pipe  or  tube.     You  can  imagine  it 
spinning  on  an  axis.     When  rolled  on  the  ground  it  runs 
in  a  straight  line.      The  complete   surface  of  the  cylinder 
consists  of  two  circular  plane  surfaces  separated  by  a  curved 
surface  (fig.  69,  ii). 

22.  The  cone  reminds  you  of  the  old-fashioned  candle- 


*  It  is  convenient  to  be  able  to  refer  to  the  "  centre  "  of  a  square,  but  it  is  not 
strictly  correct.     A  circle  has  a  true  centre,  so  has  a  sphere. 


ELEMENTARY  GEOMETRY 


237 


extinguisher,  or  of  the  sugar  loaf.  It  has  a  circular  base, 
and  it  is  so  far  like  a  pyramid  that  it  has  a  vertex  over  the 
centre  of  the  base.  You  can  imagine  it  spinning  on  an  axis. 
When  rolled  on  the  ground  it  runs  round  in  a  circle.  The 
complete  surface  of  the  cone  consists  of  one  circular  plane 
surface  and  a  curved  surface  (fig.  69,  iii). 

23.  The  sphere  reminds  you  of  a  ball  of  some  kind, 
and  it  is  a  ball  which  is  perfect  in  this  way — the  point  called 
the  centre  is  exactly  the  same  distance  from  every  point 
on  the  surface.     You  can  imagine  it  spinning  on  an  axis, 
like  the  earth.     When  rolled  on  the  ground,  it  will  run  in 
any  direction.    The  surface  of  a  sphere  is  everywhere  curved 
(fig.  69,  iv). 

24.  When  we  speak  of  a  "  cylindrical  surface  "  or  of  a 
"  conical  surface  ",  we  usually  refer  to  only  the  curved  surface 
of  the  cylinder  or  cone.     It  is  important  to  notice  that  this 
curved  surface  of  the  cylinder  and  the  cone  is  very  different 
from  the  curved  surface  of  a  sphere.    If  you  place  a  sphere 
upon  a  plane  (say  a  table),  it  touches  the  plane  in  a  point. 
If  you  allow  a  cylinder  or  a  cone  to  lie  with  its  curved  surface 
on  a  plane,  it  touches  the  plane  in  a  line. 

25.  We  can  make  nets  of  a  cylinder  and  a  cone,  but  not 
of  the  sphere.    Here  arc  nets  of  a  square  pyramid,  a  cylinder, 


and  a  cone,  but  to  make  models  from  the  nets  of  the  last 
two  is  a  little  difficult. 

26.  You  have  learnt  to  recognize  a  square,  an  oblong, 
a  triangle,  and  a  circle.  All  these  are  called  plane  figures, 


238  CRAFTSMANSHIP  IN  MATHEMATICS 

because  each  encloses,  within  a  boundary  line,  part  of  a 
plane  surface.  All  plane  figures  have  closed  boundary 
lines.  The  letter  O  and  the  letter  D  are  geometrical  figures, 
but  not  the  letter  C  or  the  letter  W.  Straight-lined  figures 
like  squares  and  triangles  are  called  rectilineal  figures. 
A  circle  is  a  curved  figure. 

27.  A   PLANE   FIGURE  is  part  of  a  Plane,  and  it 
is  separated  from  the  rest  of  the  Plane  by  a  boundary 
line.    (L.) 

28.  A  PLANE  RECTILINEAL  FIGURE  is  a  straight- 
lined  figure  on  a  Plane.    (L.) 

29.  RECTI-LINEAL  means  straight-lined.  (L.) 

LESSON  IV 
Angles 

30.  If  I  stand  facing  the  east  and  the  drill  sergeant  says 
"  left  turn  ",  I  turn  and  face  the  north,  and  I  have  then 

turned  through  a  right  angle.     If  he  repeats 
the  order,  I  turn  to  the  west,  and  I  have  then 
turned   through  another  right   angle.      If  he 
repeats  the  order  twice  more,  I  turn  and  face 
south  and  then  turn  and  face  east,  by  which 
time  I  shall  have  turned  through  four  right 
angles.     I  have  made  one  complete  rotation 
(Lat.  rota  =  a  wheel).     Note  the  little  arrow  showing   my 
first  quarter-rotation  or  right  angle. 

Evidently  an  angle  may  be  smaller  or  greater  than  a  right 
angle.  Whenever  you  look  at  a  clock,  the  two  hands  are 
making  an  angle  with  each  other.  In  fact,  they  are  making 
new  angles  with  each  other  all  day  long.  Even  when  they 
are  exactly  together  they  have  just  completed  a  new  angle 
and  are  just  beginning  to  make  others. 

31.  On  paper  an  angle  is  represented  by  two  lines  meeting 
in  a  point.    The  two  lines  are  called  the  arms  of  the  angle, 
and  the  point  where  the  two  arms  meet  is  called  the  vertex 


w 


ELEMENTARY  GEOMETRY  239 

of  the  angle.  The  same  angle  may  have  long  arms  or  short 
arms.  If  a  big  clock  and  a  little  watch  are  both  keeping 
correct  time,  the  angles  between  their  hands  are  always 
exactly  the  same.  An  angle  always  represents  an  amount 
of  movement,  namely,  the  movement  of  rotation.  One 
arm  shows  where  the  rotation  began,  and  the  other  where  it 
finished,  and  you  must  always  think  of  an  angle  in  this  way. 

32.  You  can  measure  angles  of  different  sizes  fairly  well 
by  opening  and  closing  your  dividers,  but  the  joint  prevents 
you  from  making  an  angle  of  a  whole  rotation.  A  more 
convenient  form  of  angle -measurer  is  necessary,  and  you 
may  make  one  in  this  way.  Take  two  nicely  planed  strips 
of  wood,  say  about  12"  long,  |"  wide,  -J"  thick,  and  pivot 
them  together,  something  like  your  dividers,  by  means  of 
a  tiny  brass  bolt  with  rounded  head  and  nut,  generally 
obtainable  for  a  penny  or  two  from  the  ironmonger's.  If 
you  cannot  obtain  these  things,  two  strips  of  cardboard 
will  do,  pivoted  on  a  long  drawing-pin,  head  downwards, 
with  a  protecting  bit  of  cork  over  the  point. 

Place  your  angle-measurer  on  the  table  before  you,  the 
vertex  O  to  the  left,  the  two  arms  OA,  OB  together  as  if 


A,B 


fcj 

.L, 


Fig.  72 


they  were  both  pointing  to  III  on  a  clock-face  (fig.  72,  i). 
Keep  the  under  arm  OA  fixed,  and  rotate  the  upper  arm 
OB.  Rotate  it  in  an  anti-clockwise  direction  (this  is  the  custom 
in  geometry),  and  make  angles  equal  to  one,  two,  three,  and 
four  right  angles.  Draw  the  four  angles,  and  in  each  case 
show  the  amount  of  rotation  by  means  of  little  curved  arrows 
(fig.  72,  ii,  iii,  iv,  v). 

33.  In  measuring  different  quantities,  weights  and  mea- 
sures for  instance,  big  units  like  tons  and  miles  are  some- 


240  CRAFTSMANSHIP  IN  MATHEMATICS 

times  inconvenient.  We  do  not  weigh  our  tea  in  tons  or 
measure  our  pencils  in  miles;  we  use  smaller  units  like 
ounces  and  inches.  So  with  angles.  A  right  angle  is  a  rather 
00*  big  unit,  and  sometimes  we  use  a 

smaller  unit  called  a  degree.  If  we 
make  a  right  angle  as  before,  but  move 
OB  into  position  gradually,  in  ninety 
equal  steps,  each  of  these  steps  is  an 
angle  of  one  degree.  It  is  a  very 
small  angle,  too  small  to  be  shown 
clearly  on  paper  unless  we  make  the 
ng,  73  arms  very  long.  The  figure  shows  that 

even  an  angle  of  5  degrees  is  very  small. 
The  sign  for  "  degree  "  is  a  little  circle  placed  at  the  top 
right-hand   corner   of  the   given   number.      Thus   for   "  35 
degrees  "  we  write,  "  35°  ". 

34.  We  may  make  up  a  little  table: 

90  degrees  make  a  right  angle, 

2  right  angles  make  a  straight  angle, 

2  straight  angles  make  a  perigon. 

A  perigon  is  an  angle  of  one  complete  rotation  (pert  —  round; 
gon  ~  angle).  It  is  equal  to  four  right  angles,  or  360°.  A 
straight  angle  contains  180°  (§  32,  fig.  72,  iii). 

We  choose  the  number  360  for  the  perigon  simply  because 
it  is  the  number  which  contains  many  useful  factors  (2,  3, 
4,  5,  6,  8,  9,  10,  12,  15,  18,  20,  24,  30,  36,  40,  45,  60,  72, 
90,  120,  180).  Any  other  number  would  do,  but  it  would  be 
less  useful.  The  French  use  the  number  400;  they  prefer 
to  divide  up  the  right  angle  into  100  degrees  (they  call  them 
grades)  instead  of  90. 

35.  Note  the  number  of  degrees  in  the  angles  of  fig.  74. 
The  dotted  lines  show  the  right  angles,  and  help  the  eye 
to  estimate  the  numbers  of  degrees. 

Practise  drawing  angles  of  different  sizes,  and  estimating 
the  number  of  degrees.  The  most  important  angles  of  all 
are  30°,  45°,  60°,  90°,  180°.  The  easiest  to  make  is,  of  course 


ELEMENTARY  GEOMETRY 


241 


an  angle  of  90°.    Divide  it  into  two  equal  parts  as  accurately 
as  you  can,  and  so  obtain  45°.     It  is  pretty  easy  to  divide, 


iB 


Fig.  74 


with  fair  accuracy,  45°  into  three  equal  parts,  in  order  to 
obtain  15°  and  30°.    And  so  on. 

But  guesswork  will  certainly  not  always  do.  It  is  often 
necessary  to  draw  given  angles  accurately,  and  for  this  pur- 
pose you  must  use  a  pro- 
tractor. A  protractor  is 
a  semicircular  *  piece  of 
brass  or  celluloid,  with 
numbers  from  0°  to  180° 
round  the  circumference, 
in  both  directions,  and  by 
means  of  it  you  can  make  Fig.  75 

an  angle  of  any  size. 

Suppose  you  have  to  draw  a  line,  at  say,  55°  with  a  given 
line.  Place  the  straight  diameter  of  your  protractor  against 
the  given  line,  in  such  a  way  that  the  marked  midpoint 


*A  surveyor's  protractor  is  circular,  and  is  numbered  up  t 
geometry  we  do  not  very  often  require  angles  greater  than  ioo°. 

(B291) 


to  360°.     But  in 
17 


242  CRAFTSMANSHIP  IN  MATHEMATICS 

of  the  diameter  is  against  that  end  of  the  line  that  is 
to  be  the  vertex  of  the  angle.  At  the  number  55  on  the 
circumference,  mark  a  point  on  the  paper.  Remove  the 
protractor,  and  through  that  point  draw  the  second  arm  of 
the  angle. 

But,  you  will  say,  there  are  two  55's  on  the  circumference. 
How  are  we  to  choose  between  them?  That  is  easy,  for 
you  know  that  55°  is  less  than  a  right  angle,  and  you  choose 
the  55  which  will  give  you  such  an  angle.  The  other  55 
would  be  used  if  the  vertex  of  the  angle  had  to  be  at  the 
other  end  of  the  line. 

36.  When  two  straight  lines  stretch  out  from  one 
point,  like  two  spokes  from  the  hub  of  a  wheel,  they 
form  an  angle.    (L.) 

37.  The  two  lines  are  called  the  arms  of  the  angle, 
and  the  point  where  they  meet  is  called  the  vertex 
of  the  angle.    (L.) 

38.  An  angle  always  shows  a  certain  amount  of 
ROTATION    round    the    vertex,    one    arm    showing 
where  the  rotation  began,  the  other  arm  showing  where 
it  ended.   (L.) 

39.  A    PERIGON   is   an   angle   of  one   complete 
rotation.    (L.) 

40.  A    STRAIGHT   ANGLE  is  an  angle  of  a  half 
rotation.   (L.) 

41.  A    RIGHT    ANGLE  is  an  angle  of  a  quarter 
rotation.    (L.) 

42.  AN  ANGLE  OF  ONE  DEGREE  is  an  angle  of 
sio  Part  °*  a  rotation. 

LESSON  V 
Surveyors  and  their  Work 

43.  A  surveyor's  work  is  to  measure  up  land,  and  to 
draw  plans  and  maps.     For  measuring  lengths,  he  uses  a 
long  chain  of  100  links.    For  measuring  angles,  he  uses  an 


ELEMENTARY  GEOMETRY  243 

angle-measurer  which  is  like  yours  in  this  respect — that 
it  consists  of  two  pivoted  arms;  but  it  is  much  more  elaborate 
than  yours,  for  he  has  to  measure  angles  very  accurately. 
He  also  uses  a  levelling-staff,  to  help  him  measure  differences 
of  level.  A  levelling-staff  is  merely  a  pole,  graduated  to  show 
heights  above  the  ground. 

44.  Here  is  a  problem  in  which  the  necessary  angle 
measurements  may  be  correctly  and  easily  made  with  one 
of  your  set-squares.  To  solve  it  you  must  make  a  drawing 
to  scale. 

A  and  B  are  two  towns  20  miles  apart.     Another  town  C 
is  60°  east  of  north  from  A  and  30° 
west  of  north  from  B.    Draw  a  plan  to 
show  the  position  of  C,  and  give  its 
distances  from  A  and  B. 

The  line  AB  is  20  miles  long, 
and  we  have  to  draw  it  to  a  suitable 
scale.  A  scale  of  -|"  to  the  mile  would 
do.  Thus  we  make  AB  20  eighth-inches,  or  2|",  long. 

If  C  were  exactly  north  of  A,  it  would  be  somewhere 
in  the  line  AD.  But  it  is  60°  east  of  this  line,  and  we  there- 
fore make  the  angle  DAF  equal  to  60°.  Again,  if  C  were 
exactly  north  of  B,  it  would  be  somewhere  in  the  line  BE. 
But  it  is  30°  west  of  this  line,  and  we  therefore  make  the 
angle  EBG  equal  to  30°.  We  know  now  that  the  town  C 
lies  on  both  AF  and  BG.  But  the  only  place  where  it  can 
lie  on  both  is  where  they  meet.  Hence,  mark  this  point,  C. 
We  have  thus  found  the  position  of  C. 

To  find  the  distances  CA  and  CB,  we  measure  them  to 
scale.  CA  is  nearly  17J  eighth-inches  long,  and  CB  is  10 
eighth-inches  long.  Thus  C  is  17£  miles  from  A  and  10 
miles  from  B. 

But  this  problem  was  a  problem  on  paper.  No  part  of 
the  work  was  done  with  measuring  instruments  in  the 
field.  Let  us  come  back  to  the  surveyor. 

Sometimes  a  surveyor  works  on  level  ground,  and  has  to 
measure  angles  in  a  horizontal  plane.  Sometimes  he  works 


244 


CRAFTSMANSHIP  IN  MATHEMATICS 


on  hilly  ground  and  has  to  measure  angles  in  a  vertical  plane. 

45.  Measuring  an  angle  in  a  horizontal  plane. — 

Suppose  you  are  standing  at  a  place  P  in  a  field,  and  you 
imagine  a  line  drawn  from  your  eye  to  each  of  two  distant 
trees,  Tl  and  T2.  What  is  the  angle  between  the  lines?  Set 
up  a  table  at  P,  with  a  piece  of  drawing-paper  pinned  on 
it.  (A  camera  tripod  stand  with  a  drawing-board  fixed  on  it 
horizontally  about  the  height  of  your  top  waistcoat  button 
would  do  nicely.)  Place  an  angle-measurer  on  the  table, 
swing  one  arm  round  to  point  to  Tx,  and  the  other  round 
to  point  to  T2.  Hold  the  arms  firmly  and  draw  the  two 
angle  lines  (against  the  inside  of  the  arms),  remove  the  measurer, 
and  with  your  protractor  find  the  number  of  degrees  in  the 
angle. 

(Ask  the  geography  master  to  show  you  his  plane-table 
and  to  explain  how  he  measures  angles  made  by  distant 
objects.  With  his  angle-measurer  pivoted  to  the  centre  of  a 
circular  protractor  on  the  table,  he  is  able  to  read  at  once 
any  angle  made  by  the  two  arms.) 

46.  Measuring  an  angle  in  a  vertical  plane. — Pivot 
your  angle-measurer  to  the  side  of  a  short  post,  or  to  the 

side  of  a  stout  stick  thrust 
vertically  into  the  ground,  in 
order  that  the  arms  may  swing 
in  a  vertical  plane.  An  ordi- 
nary drawing-pin  makes  a  poor 
pivot,  for  it  is  then  difficult  to 
make  the  arms  remain  in  a 
particular  position.  An  angle- 
measurer  made  of  wood,  with 
a  fairly  tight  wooden  or  metal 
pivot,  is  much  more  satisfactory 
than  the  pivoted  cardboard 
strips.  An  angle  measured 

in  a  vertical  plane  is  always  an  angle  with  a  hori- 
zontal arm;  the  other  arm  points  upwards  or  downwards 
as  may  be  necessary. 


Angle  of  elevation        Angle  of  depression 
Fig.  77 


ELEMENTARY  GEOMETRY  245 

If  you  are  on  low  ground  and  want  to  measure  the  angle 
made  by,  say,  a  cottage  at  the  top  of  the  hill,  point  the  one 
arm  of  your  angle-measurer  upwards  to  the  cottage,  and 
measure  the  angle  of  elevation.  If  you  are  on  high  ground, 
say  the  top  of  a  cliff,  and  want  to  measure  the  angle  made 
by  a  boat  in  the  water  below,  point  the  one  arm  of  your 
angle-measurer  downwards  to  the  boat,  and  measure  the 
angle  of  depression.  Since,  for  measuring  different  angles, 
the  arms  may  have  to  swing  round  in  different  vertical  planes, 
it  is  an  advantage  to  be  able  to  turn  the  post  round  in  the 
ground,  and  it  should  therefore  have  a  rounded  point,  some- 
thing like  the  point  of  a  cricket-stump,  prolonged. 

(Ask  the  geography  master  to  show  you  his  clinometer,  and 
to  explain  how  he  reads,  from  the  cardboard  protractor, 
angles  of  elevation  and  depression.  Try  to  understand  the 
use  of  the  little  plumb-line,  and  observe  the  pivot  on  which 
the  protractor  turns.) 

47.  How  can  I  find  the  width  of  a  river  which  I 
cannot  cross? — To  solve  this  problem  we  have  to  measure 
angles  in  a  horizontal  plane.  E 

Let  AB  and  CD  repre-       A  ~° 

sent  the  two  banks.     I  note 

some  object  E  on  the  oppo-      C 

site   bank,   and    I    measure  Fig.  78 

any    length    FG,    say    100 

yards,  on  the  near  bank.  Then  I  measure  the  horizontal 
angles  at  F  and  G,  in  each  case  pointing  one  arm  of  my  angle- 
measurer  along  CD  and  the  other  arm  to  the  object  E.  I 
note  that  angle  EFG  =  60°,  and  angle  EGF  =  45°.  Now  I 
am  ready  to  make  a  drawing  to  scale.  A  scale  of  1"  to 
50  yards  seems  convenient,  so  that  FG  =  2".  The  width 
of  the  river  is  shown  by  a  perpendicular  EH  drawn  from 
E  to  FG.  Measuring  EH  to  scale,  I  find  it  is  very  nearly 
yards. 

48.  How  can  I  find  the  height  of  the  flagstaff  in  the 
school  field? — To  solve  this  problem  I  have  to  measure  an 
angle  in  a  vertical  plane,  and  as  one  arm  of  my  angle-measurer 


246 


CRAFTSMANSHIP  IN  MATHEMATICS 


will  have  to  point  upwards,  the  angle  will  be  an  angle  of 
elevation. 

Let  AB  be  the  flagstaff,  and  let  CD  be  the  post  to  which 
my  angle-measurer  is  attached:  a  convenient  height  of  this 
attachment  is  4'  above  the  ground.  The  post  may  be  fixed 
at  any  measured  distance  from  the 
flagstaff,  say  20'.  Thus  DB  =  20'. 
The  horizontal  arm  of  the  angle- 
measurer  points  to  E  in  the  flagstaff; 
E  is  therefore  4'  above  the  ground. 
The  other  arm  points  to  the  top  of 
the  flagstaff.  I  now  measure  the  angle 
ACE,  and  find  it  is  60°.  Now  I  am 
ready  to  make  a  drawing  to  scale,  say 
1"  to  10',  so  that  CE  (=  DB)  =  2", 
angle  ACE  -  60°,  angle  AEC  =  90°. 
The  length  of  AE,  measured  to  scale, 
is  34-6'.  Hence  AB  =  34-6'  +  4' 
=  38-6'. 

(The  length  BE  is  exaggerated  in 
the  printed  figure.) 

49.  The  next  time  you  see  a  surveyor  at  work,  ask  him 
to  show  you  the  instruments  he  uses  for  measuring  horizontal 
and  vertical  angles,  and  to  explain  how  he  is  able  to  measure 
even  very  small  fractions  of  a  degree.  Also  ask  him  to  tell 
you  something  about  his  levelling-staff  and  his  chain. 


LESSON  VI 
Symmetry 

50.  Stand  in  front  of  a  looking-glass,  with  a  book  (or 
some  other  object)  in  your  right  hand.  In  the  glass  you  see 
an  image  of  yourself,  but  the  image  holds  the  book  in  his  left 
hand.  Close  your  left  eye;  the  image  closes  his  right  eye. 

Hold  open  your  right  hand  in  front  of  the  glass,  and  look 
at  the  image  of  the  palm.  Compare  this  image  with  the 


ELEMENTARY  GEOMETRY  247 

palm  of  your  real  left  hand.     They  are  exactly  alike.     For 
instance,  the  two  thumbs  point  in  the  same  direction. 

Thus  the  image  of  your  right  hand  is  a  left  hand. 
In  short,  your  two  hands  are  not  in  all  respects  alike;  each 
is  the  "  image  "  of  the  other. 

51.  Place   a   pair  of  gloves   side   by  side  on   the  table, 
backs  upwards,  thumbs  touching.    Each  is  the  image  of  the 
other.    Turn  the  left  glove  inside  out;   it  has  become  a  right- 
hand  glove.    You  now  have  two  right-hand  gloves,  no  longer 
images  of  each  other  but  like  each  other. 

52.  Fold  a  sheet  of  white  paper,  like  a  sheet  of  note- 
paper,  and  smooth  down  the  crease.     Open  again,  and  let 
a  drop  of  ink  fall  in  the  crease.     Now  fold,  and  press  the 
folded  paper  fairly  hard,  to  make  the  ink  run  and  form  a 
pattern.     Open;    the  two  half-patterns  are  right-  and  left- 
handed;    each  is  the  image  of  the  other.     When  the  paper 
is  folded  on  its  crease  and  held  up  to  the  light,  the  two  half- 
patterns  are  seen  to  fit  over  each  other  exactly. 

53.  Right-  and  left-handed   patterns  that  can  be  folded 
exactly  together  in  this  way,  and  are  thus  images  of  each 
other,  are  said  to  be  symmetrical.    The  dividing  line  re- 
presented by  the  crease  is  called  the  axis  of  symmetry. 
We  say  that  the  doubled  pattern  is  symmetrical  with  respect 
to  the  axis. 

54.  Take   another  sheet   of  paper,  and  fold   as  before. 
Let  a  drop  of  ink  fall  inside,  but  at  some  distance  from  the 
crease.     Press  down  the  doubled  paper,  and 

so  form  ink  figures.  The  figures  are  images  of 
each  other  and  fold  together  exactly  as  before, 
but  this  time  they  do  not  touch  the  crease  (the 
axis).  That  does  not  matter.  They  are  still 
symmetrical  with  respect  to  the  axis. 

55.  In  the  accompanying  figure  (a  kite), 

the  line  AB  is  evidently  an  axis  of  symmetry,  FI«.  so 

for  the  half  ACB  can  be  folded  over  on  AB 
and  be  made  to  fit  exactly  on  the  other  half.     The  one  half 
is  the  image  of  the  other.    Hence  corresponding  lines  on 


248  CRAFTSMANSHIP  IN  MATHEMATICS 

the  left  and  right  must  be  equal  in  length.  If  we  hold 
the  doubled  paper  up  to  the  light,  we  can  see  that  the  lines 
are  equal.  Corresponding  angles  must  also  be  equal. 

56.  ABC    and    DBF  are  two  figures   symmetrical  with 
respect  to  the  axis  MN.     Hence,  if  we  fold  on  MN,  the 

figures  will  fit  together  exactly.  On 
ABC,  mark  the  two  points  G  and 
H,  fold  over,  and  prick  through  G 
and  H,  on  DEF.  On  opening  out 
we  shall  find  the  two  points  K  and 
L  in  positions  corresponding  ex- 
actly to  G  and  H,  and  KL  is 
evidently  equal  to  GH. 

57.  Every  point,  every  line,  every 
Fig.  si  angle,  in    one    of    two    symmetrical 

figures   has  an  image    in  the  other. 

The  image  always  corresponds  exactly  to  the  original.  The 
two  have  exactly  corresponding  positions. 

58.  Thus   a  point  and  its  image  are  always  equidistant 

from  the  axis.    In  the  last  figure,  for  instance,  A  and  D  are 

equidistant   from   MN,   for   they   come   together    when   the 

figures  are  folded  about  the  axis.    Hence  if  we  join  AD,  the 

axis  must  bisect  AD. 

59.  Fold  a  piece  of  paper  and   mark  the 
crease  as  an  axis  AB.     On  one  side  of  the  axis, 
make  a  point  M.     Fold,  and  prick  through  M 
to  obtain  its  image  N.     Join  MN.     PM  =  PN 
(by   §   58).      Angle   APM  =  angle  APN   (by 
§  57).     But  angle  MPN  is  a  straight  angle,  and 
thus  the  equal  angles  APM  and  APN  are  both 
right  angles.    Also,  the  angles  vertically  opposite 
Fig.  82          these  are  equal.    Hence  all  four  angles  at  P  are 
right  angles.     We  see  now  that  the  axis  not 
only  bisects  MN  but  is  perpendicular  to  it,  that  is,  the  axis 
is   the   perpendicular    bisector   of  MN.     Observe   that 
whenever  you  fold  a  sheet  of  paper  a  second  time,  as  when 
you  put  it  into  an  envelope,  you  make  two  axes  of  symmetry 


M 


ELEMENTARY  GEOMETRY 


249 


perpendicular  to  each  other,  the  four  perfect  right  angles 
fitting  exactly  together  in  the  envelope.  When  the  paper 
is  opened  out,  you  see  the  complete  perigon  they  form. 

60.  Fold  a  piece  of  paper,  and  then  fold  a  second  time, 
thus  obtaining  two  axes  of  symmetry,   MN  and  PQ,  and 
four  right  angles.     The  four  divisions  are  sometimes  called 
quadrants.    Prick  through  all  four  thicknesses  of  the  folded 
paper,  in  four  or  five  points  not  in  the  same  straight  line. 
Open    out,    and    join    up    the 

points,  in  the  same  manner,  in 
the  four  quadrants,  thus  mak- 
ing four  figures.  Convince 
yourself  that  both  MN  and  PQ 
are  really  axes,  by  first  folding 
on  MN,  holding  up  to  the  light 
and  seeing  that  the  left  and 
right  halves  of  the  whole  fit, 
then  folding  on  PQ  and  seeing 
that  the  upper  and  lower  halves 

fit.  Join  any  point,  say  A,  to  its  image  on  the  other  side 
of  each  axis,  namely  to  B  on  the  other  side  of  MN,  and  to 
C  on  the  other  side  of  PQ.  Observe  that  the  axes  are  the 
perpendicular  bisectors  of  the  respective  joining  lines,  MN 
of  AB,  and  PQ  of  AC.  So  it  is  generally. 

61.  When  two  figures  are  symmetrical  with  respect 
to  an  axis,  they  are  right-  and  left-handed,  and  when 
they  are  folded  about  the  axis,  they  fit  together  exactly. 
(L.) 

62.  An    axis   of   symmetry   is    the    perpendicular 
bisector  of  the  line  joining  any  point  on  one  side  of 
the  axis  to  its  image  on  the  other  side. 


250  CRAFTSMANSHIP  IN  MATHEMATICS 

LESSON  VII 
Congruent,  Symmetrical,  Similar 

63.  We  often  require  a  word  to  describe  two  figures 
which  are  alike  in  all  respects — corresponding  lines  the 
same  length,  corresponding  angles  the  same,  areas  the  same, 
appearances  the  same. 

When  two  figures  are  exactly  alike  in  all  respects,  and 
can  be  made  to  fit  exactly  together,  they  are  said  to  be  con- 
gruent. (Congruent  means  exactly  agreeing.)  Here  are 
three  pairs  of  congruent  figures. 


Fig.  84 

64.  Symmetrical  figures  are  exactly  alike  in  all  respects 
save  one:  they  are  right-  and  left-handed.  To  make  two 
symmetrical  figures  fit  exactly  together,  we  have  to  turn 
one  of  them  over  through  a  straight  angle  (180°),  round 
the  axis  of  symmetry.  It  is  like  picking  one  up,  turning  it 
upside  down,  and  putting  it  down  again.  Then  the  two 
will  fit  exactly. 

Strictly  speaking,  we  ought  not  to  call  symmetrical 
figures  congruent,  because  they  are  not  alike  in  all  respects; 
they  are  right-  and  left-handed.  But  it  has  become  customary 
to  call  even  symmetrical  figures  congruent,  because  they  can 
be  made  to  fit  exactly  if  one  is  turned  over. 


i.  Congruent  ii.  Symmetrical  iii.  Similar 

(but  not  symmetrical)  (and  congruent)  (neither  congruent  nor 

symmetrical) 
Fig.  85 

65.  But  similar  is  another  term   altogether.     Similar 


ELEMENTARY  GEOMETRY  251 

figures  are  figures  of  the  same  appearance,  irrespective 
of  their  size.     (See  fig.  85.) 

66.  Congruent  figures  which  are  not  symmetrical  may 
be  made  to  fit  together  exactly  by  sliding  one  over  the  other. 
But  symmetrical  figures  cannot  be  made  to  fit  by  sliding; 
one  has  first  to  be  turned  over. 

You  might  say  that  since  congruent  figures  are  alike 
in  appearance,  we  might  call  them  similar.  That  is  true, 
but  in  geometry  we  do  not  usually  apply  the  term  similar 
to  congruent  figures  unless  they  are  of  different  sizes. 

67.  Notice    two    important    things    about    similar 
figures:    (1)  all  the  angles  in  the  one  are  equal  to  the 
corresponding  angles  in  the  other;    (2)  the  proportions  in 
the  one  are  equal  to  the  proportions  in  the  other.     (If,  for 
instance,  the  big  pig's  tail  is  one-third  the  length  of  his 
back,  the  little  pig's  tail  is  one-third  the  length  of  his  back.) 
You  will  learn  more  about  "  proportions  "  later  on. 

68.  CONGRUENT  figures  are  figures  exactly  alike 
in  all  respects.     One  can  be  made  to  slide  over  the 
other  and  fit.    (L.) 

69.  SYMMETRICAL   figures  are  right-  and  left- 
handed  congruent  figures.  To  make  them  fit,  one  has 
to  be  turned  over  through  180°,    (L.) 

70.  SIMILAR  figures  are  figures  of  different  sizes, 
but  they  have  the  same  appearance,  the  same  pro- 
portions, and  the  same  angles.    (L.) 

LESSON  VIII 
Classifying  and  Defining 

71.  When  we  arrange  a  number  of  things  in  separate 
classes,  we  are  said  to  classify  them. 

We  may,  for  instance,  arrange  all  school  exercise-books 
in  two  quite  distinct  classes,  namely,  ruled  and  unruled. 
Such  a  classification  is  good.  But  suppose  we  say  that 
all  the  people  in  London  are  either  males,  or  females,  or 


252  CRAFTSMANSHIP  IN  MATHEMATICS 

Australians.    The  classification  is  bad,  for  the  Australians  have 
been  included  twice  over;   they  are  all  males  or  females. 

Here  is  another  example  of  a  good  classification.  In  a 
certain  school,  the  100  boys  in  Form  IV  are  grouped  in 
four  divisions,  according  to  the  languages  they  learn  in 
addition  to  English  and  French. 

Form  IVa  learn  both  Latin  and  Greek,  but  not  German. 
Form  IVb  learn  both  Latin  and  German,  but  not  Greek. 
Form  IVc  learn  Latin,  but  not  Greek  or  German. 
Form  IVd  learn  German,  but  not  Latin  or  Greek. 

There  are  four  distinct  divisions.      Every  boy  is  included 
once,  and  only  once. 

72.  Now  we  will  classify  triangles.  All  triangles  are 
either  isosceles  or  scalene.  But  isosceles  triangles  are  of  two 
kinds,  those  with  two  sides  equal,  those  with  all  three  sides 
equal.  Thus  we  may  arrange  the  classes  in  this  way: 

Triangles 


no  two  sides  equal  at  least  two  sides  equal 

I 

I  I 

only  two  sides  equal     all  three  sides  equal 
or  we  may  arrange  in  this  way: 
Triangles 


scalene  isosceles 


I  I 

base  shorter  or  longer        base  equal  to  the  equal  sides 
than  the  equal  sides  equilateral 

73.  Defining. — A  thing  (it  may  be  a  dog  or  it  may 
be  a  triangle)  has  a  name,  and  that  name  is  a  word.  In 
order  to  say  what  that  word  means,  we  have  to  make  a 
short  statement  which  will  show  how  the  thing  is  dis- 


ELEMENTARY  GEOMETRY  253 

tinguished  from  all  other  things.  That  short  statement  is  a 
definition.  We  define  a  word,  and  the  definition  must 
include  the  leading  property  of  the  thing. 

We  begin  by  thinking  of  the  class  to  which  the  thing 
belongs.  Suppose,  for  instance,  we  have  to  define  a  chair. 
To  what  class  of  things  does  a  chair  belong?  Evidently 
to  the  class  articles  of  furniture.  Thus  we  may  begin 
by  saying, 

A  chair  is  an  article  of  furniture  .  .  . 

Now  we  have  to  pick  out  the  particular  property  which 
distinguishes  a  chair  from  all  other  articles  of  furniture. 
What  is  a  chair  specially  used  for?  For  sitting  on.  Thus 
we  may  now  say, 

A  chair  is  an  article  of  furniture  for  sitting  on. 

But  benches,  sofas,  and  stools  are  also  used  for  sitting  on. 
How  are  we  to  distinguish  a  chair  from  these?  Benches 
and  sofas  are  made  for  more  than  one  person  to  sit  on.  So 
we  may  say, 

A  chair  is  an  article  of  furniture  for  one  person  to  sit  on. 

But  this  might  apply  to  a  stool.  How  are  we  to  distinguish? 
A  chair  has  a  back,  a  stool  has  not.  We  therefore  say, 

A  chair  is  an  article  of  furniture  for  one  person  to  sit  on 
and  to  lean  back  against. 

Again,  define  a  pair  of  compasses.  To  what  class 
does  it  belong?  Mathematical  instruments.  What  is  its  special 
use?  For  drawing  circles.  Thus  we  make  up  the  definition: 

A  pair  of  compasses  is  a  mathematical  instrument  for  drawing 
circles. 

74.  Define  a  triangle. 

To  what  class  does  it  belong?    Plane  rectilineal  figures. 
What    property    distinguishes    it    from    all    other    plane 
rectilineal  figures?    It  has  three  sides.    Therefore  we  say, 

A  triangle  is  a  plane  rectilineal  figure  with  three  sides. 


254  CRAFTSMANSHIP  IN  MATHEMATICS 

Define  an  isosceles  triangle. 

To  what  class  does  it  belong?   Triangles. 

What  distinguishes  isosceles  triangles  from  the  other 
great  class  of  triangles  (scalene)?  Equality  of  the  two  sides 
from  the  vertex  to  the  base. 

Therefore  we  say, 

An  isosceles  triangle  is  a  triangle  in  which  the  two  sides 
from  the  vertex  to  the  base  are  equal. 

Define  an  equilateral  triangle. 

To  what  class  does  it. belong?    Isosceles  triangles. 
What  distinguishes  it  from  other  isosceles  triangles?    The 
base  is  equal  to  each  of  the  other  two  sides. 
Therefore  we  say, 

An  equilateral  triangle  is  an  isosceles  triangle  in  which 
the  base  is  equal  to  each  of  the  other  two  sides. 

Another  definition  of  an  equilateral  triangle  is  some- 
times given:  an  equilateral  triangle  is  a  triangle  with  three 
equal  sides.  But  this  definition  is  not  so  good  as  the  other. 

75.  We  might,  if  we  liked,  classify  triangles  according 
to  their  angles,  and  ignore  their  sides.  The  sum  of  the  three 
angles  of  a  triangle  is  180°.  Hence,  if  a  triangle  has  an  obtuse 
angle,  the  other  two  angles  must  be  acute;  or  if  it  has  a  right 
angle,  the  other  two  angles  must  be  acute;  if  it  has  neither 
an  obtuse  angle  nor  a  right  angle,  all  three  angles  must  be 
acute.  Thus  we  have  a  new  classification:  All  triangles 
are  either  obtuse-angled  triangles,  or  right-angled 
triangles,  or  acute -angled  triangles. 

But  do  not  mix  up  the  two  classifications  of  triangles. 
That  would  take  us  back  to  the  Australians! 

Such  lessons  are  easily  within  the  range  of  very  young 
boys. 

Some  teachers  are,  however,  curiously  afraid  of  the  prin- 
ciple of  symmetry,  urging  that  it  does  not  lend  itself  to 


ELEMENTARY  GEOMETRY  255 

strictly  deductive  proof.  Personally  I  would  use  it  very 
much  more  for  teaching  even  advanced  geometry;  I  always 
did  in  my  teaching  days.  For  elementary  work  at  all  events, 
it  is  a  singularly  useful  weapon.  Though  proof  by  means 
of  it  is  difficult  for  beginners  to  set  out,  it  produces  con- 
viction in  the  beginner,  a  great  gain. 

It  will  be  observed  that,  for  framing  definitions,  we  have 
used  the  old  device  per  genus  et  differentiam.  This  is  probably 
the  only  safe  method  for  beginners.  From  schoolboys  we 
must  be  satisfied  with  something  much  less  than  perfection 
in  their  definitions.  In  particular,  do  not  worry  about  "  re- 
dundant "  definitions.  In  the  early  stages  they  are  inevitable; 
they  are  then  almost  to  be  encouraged.  It  is  much  better 
to  let  a  young  boy  say  that  "  a  rectangle  is  a  right-angled 
parallelogram  "  than  "  a  rectangle  is  a  parallelogram  with 
a  right  angle  ".  A  beginner  naturally,.,  regards  the  latter 
with  suspicion.  It  is  doubtful  wisdom  ever  to  ask  a  boy  to 
define  a  straight  line  or  an  angle.  He  has  clear  notions  of 
these  things  already,  and  these  notions  he  cannot  express 
in  language  that  is  entirely  satisfactory.  If  a  boy  says  that 
"  a  straight  line  is  the  shortest  distance  between  two  points  ", 
strictly  the  definition  is  unacceptable,  because  of  the  vague 
term  "  distance  ".  If  he  adds  as  tested  by  a  stretched  string, 
we  should  feel  that  the  idea  in  his  mind  was  clear  and  distinct; 
and  what  more  can  we  want  from  him?  As  for  an  angle,  I 
have  often  asked  boys  for  a  definition,  not  because  I  expected 
a  satisfactory  one,  but  in  order  to  show  them  that,  whatever 
definition  they  put  forward,  it  was  open  to  criticism.  Who 
has  ever  defined  either  a  straight  line  or  an  angle  satisfactorily? 
Again:  ordinarily  we  distinguish  between  a  circle  and  its 
circumference,  and  a  useful  distinction  it  is.  And  yet  we 
all  talk  about  drawing  a  circle  to  pass  through  three  points. 
However,  matters  of  this  kind  are  not  for  beginners  but  for 
the  Sixth  Form,  which  is  the  proper  place  for  a  final  polishing 
up  of  all  such  things. 


256  CRAFTSMANSHIP  IN  MATHEMATICS 

Working  Tools  for  Future  Deductive  Treatment 

These  consist  of  the  familiar  propositions  concerning: 

1.  Angles  at  a  point. 

2.  Parallels. 

3.  Congruency. 

4.  Pythagoras.^ 

5.  Circles;     such    properties    as    can    be    established 

from  considerations  of  symmetry. 

The  formal  proof  of  Pythagoras  is  easily  mastered  in  the 
Fourth  Form,  but  proofs  of  the  other  theorems  may  wait 
until  the  Sixth.  Meanwhile  all  the  propositions  must  be 
thoroughly  known  as  geometrical  facts,  facts  which  can  readily 
be  used  and  referred  to  in  all  subsequent  work.  Although 
formal  proofs  are  beyond  beginners,  the  probable  truth  of 
the  propositions  must  be  substantiated  in  some  way.  Justi- 
fication is  always  possible  at  this  stage,  though  rigorous  proof 
is  not.  Most  of  the  more  recent  textbooks  provide  "  practical  " 
proofs  of  a  kind  which  to  the  beginner  really  do  seem  to 
justify  the  claims  made  by  the  theorems.  Here,  little  need 
be  said  about  such  proofs. 

First  considerations  of  angles  at  a  point  naturally  arise 
when  the  nature  of  an  angle  itself  is  being  discussed.  Acute, 
obtuse,  adjacent,  reflex,  complementary,  supplementary, 
and  vertically  opposite  angles  may  all  be  brought  into  an 
early  lesson,  provided  that  the  rotational  idea  of  the  angle 
is  clearly  demonstrated.  Angles  up  to  360°  should  be  con- 
sidered from  the  first. 

Here  is  a  first  lesson  on  parallels  and  transversals. 

Parallel  Lines  and  Transversals 

76.  You  have  already  learnt  that  the  blue  lines  on  the 
pages  of  your  exercise  books  are  parallel,  that  is,  they  run 
in  the  same  direction  and  are  always  the  same  distance 
apart.  When  we  speak  of  "  distance  apart "  we  mean  the 


ELEMENTARY  GEOMETRY  257 

shortest  distance,  and  that  distance  is  represented  by  a 
perpendicular  from  one  line  to  the  other.  But  can  we  be  quite 
sure  that  a  line  which  is  perpendicular  to  one  of  the  parallel 
lines  is  also  perpendicular  to  the  other? 

77.  A  line  that  is  drawn  across  two  or  more  other  lines 
is  called  a  transversal  (trans  means  across).    Draw  a  trans- 
versal PQ  across  the  parallel   lines 

AB  and  CD,  cutting  AB  in  M  and  p\ 

CD  in  N.  A 

Imagine  a  man  to  walk  along  AB 
and,  on  reaching  M,  to  turn  to  the      c 
right  and  walk  along  MN.     He  has 
turned    through    the   angle   ax,   for  Fig.  86 

he    was    first    walking    towards    B, 

and  is  now  walking  towards  Q.  On  reaching  N,  let  him 
turn  to  the  left,  and  walk  along  ND.  He  has  now  turned 
through  the  angle  a2,  for  he  was  walking  towards  Q  and 
is  now  walking  towards  D.  But  now  that  he  is  walking 
along  ND  he  is  walking  in  the  same  direction  as  when 
he  was  walking  along  MB.  Hence  the  angle  he  turned 
through  on  reaching  N  is  equal  to  the  angle  he  turned  through 
on  reaching  M,  that  is,  the  angle  a2  is  equal  to  the  angle  av 

We  might  have  expected  this,  for  the  two  angles  ax  and 
a2  look  alike.    They  are  called  corresponding  angles. 

78.  When    a    transversal    is    drawn    across    two 
parallel   lines,    the   corresponding   angles   are   equal. 
(L.)    Hence, 

79.  A  transversal  which  is   perpendicular  to  one 
of  two  parallel  lines  is  also  perpendicular  to  the  other. 
(L.)    Conversely, 

80.  If    two    lines    are    both    perpendicular    to    a 
transversal,  they  are  parallel  to 

each  other.    (L.)  A 

81.  Just  as  we  showed  that  the 
corresponding  angles  c^  and  a2  are      c 
equal,    so    we   may   show   that   the 
corresponding  angles  fa  and  fa  are  Fig.  87 

(E291)  18 


258  CRAFTSMANSHIP  IN  MATHEMATICS 

equal;  also  a3  and  a4;  also  j83  and  /?4.    (Fig.  87.)    But  a2  and  a4 
are  also  equal,  because  they  are  vertically  opposite  angles. 

Since  at  =  a2,         (§  78) 

and  since          a2  =  a4, 
therefore          aA  =  a4. 

The  angles  aj  and  a4  are  on  opposite  sides  of  the  transversal, 
and  are  called  alternate  angles. 

Similarly  it  can  be  shown  that  the  alternate  angles  fi3 
and  /92  are  equal. 

82.  When    a    transversal    is    drawn    across    two 
parallel  lines,  the  alternate  angles  are  equal.    (L.) 

83.  Observe    that,   in    the  eight    marked    angles   of  the 
last  figure,  there  are  four  pairs  of  opposite  angles,  four  pairs 
of  corresponding  angles,  two  pairs  of  alternate  angles,  every 
pair  being  equal.     The  four  a's  are  equal,  and  the  four  fi's 
are  equal. 

The  four  angles  between  the  parallel  lines  are  called 
interior  angles. 

The  four  angles  outside  the  parallel  lines  are  called 
exterior  angles. 

The  following  is  very  important: 

ai  +  Pa     —  a  straight  angle, 
—  two  right  angles. 

But  p3    -  p2,  (§  82) 

therefore          ^1+^2    —  two  right  angles. 

Similarly  we  may  show  that  a4  +  /?3  =  two  right  angles. 

84.  When    a    transversal    is    drawn    across    two 
parallel  lines,  the  two  interior  angles  on  the  same 
side  of  it  are  together  equal  to  two  right  angles,  that 
is,  they  are  supplementary.   (L.) 

Considerations  of  congruency  are  best  led  up  to  by  actual 
practical  work  on  the  construction  of  triangles  from  given 
data.  One  lesson  is  enough  for  the  boys  to  discover  that  a 
triangle  can  be  described  if 


ELEMENTARY  GEOMETRY 


259 


(1)  the  3  sides  are  given, 

(2)  2  sides  and  the  included  angle  are  given, 

(3)  1  side  and  2  angles  are  given; 

md  that  therefore  two  triangles  are  congruent  if  there  is 
:orrespondence  and  equality  of 

(1)  3  sides, 

(2)  2  sides  and  the  included  angle, 

(3)  1  side  and  2  angles. 

Further  than  this  with  beginners  it  is  unnecessary  to  go. 

As  for  Pythagoras,  it  is  enough  to  give  beginners  one  or 
:wo  of  the  many  well-known  dissection  figures. 

Here  T  =  W  f  V. 


Fi«.  88 

Here  the   big  square   is   cut  up   to   form  the   two   little 
squares, 

P  =  Q  =  R  =  s  =  P'  =•.  Q'  -  R'  -  S', 
M  =  M'. 

The  fact  must  be  emphasized  that  in 
:hese  early  stages  any  attempt  at 
formal  proof  is  out  of  place.  Never- 
:heless  adequate  reasons  may  be 
xnmd,  and  should  be  provided,  in 
mpport  of  all  statements  made  con- 
cerning these  fundamental  proposi- 
ions. 

Here  is  a  lesson  on  the  centre  of  the  circle  as  a  centre 
>f  symmetry. 


260 


CRAFTSMANSHIP  IN  MATHEMATICS 


85.  The  centre  of  the  circle  as  a  centre  of  symmetry. 

— Fold  a  circle  on  a  diameter  AB  as  an  axis  of  symmetry, 
prick  through  the  two  halves  at  C,  open  out  and  call  the 
corresponding  points  Q  and  C2,  join  Ca  and  C2  and  join 
each  to  the  centre  O,  thus  forming  the  isos.  A  C1OC2. 
The  arc  of  the  sector  AQOCg  and  the  arc  of  the  segment 
ACjCg  are  the  same. 

Take  a  piece  of  celluloid  (a  piece  of  tracing-paper  will 
do,  if  you  use  it  carefully),  pin  it  down  over  the  circle  (fig. 
90,  i)  by  means  of  a  pin  thrust  through  it  and  through  the 


Fig.  90 


centre  O,  and  trace  on  the  celluloid  the  radii  OCj,  OC2, 
the  chord  QCg,  and  the  arc  C1AC2\  now  rotate  the  celluloid 
round  the  pin  (fig.  90,  ii). 

Since  the  traced  sector  and  segment  on  the  celluloid 
preserve  their  shape  and  size  while  rotating,  all  lengths 
and  angles  remain  unchanged,  and  the  arc  C^ACg  is  seen 
always  to  fit  exactly  on  the  circumference  below  it.  We  say 
that  the  rotating  sector  and  segment  are  symmetrical  with 
respect  to  the  centre  of  the  circle,  because  of  this  exact 
fitting  during  the  whole  of  a  rotation.  Thus  the  length  of 
the  arc,  the  length  of  the  chords,  the  angle  between  the 
radii,  all  remain  constant.  We  see  all  this  plainly  in  fig.  ii, 
where  the  rotating  sector  and  segment  are  shown  in  two 
positions.  Hence: 

86.  Equal  chords  in  a  circle  are  equidistant  from 
the  centre.  (L.) 


ELEMENTARY  GEOMETRY  261 

87.  Equal  chords  in  a  circle  are  subtended  by  equal 
arcs.   (L.) 

88.  Equal  angles  at  the  centre  of  a  circle  are  sub- 
tended by  equal  chords  and  by  equal  arcs.    (L.) 

89.  If  two  chords  of  a  circle  are  equal,  they  cut 
off  equal  segments.    (L.) 

90.  All    these    things    (§§    86-89)   which    apply  to  one 
circle  also  apply  to  equal  circles,  since  equal  circles  will  fit 
together  exactly. 

Early  Deductive  Treatment 

Do  not  expect  any  rigorous  logic  from  beginners.  We 
suggest  a  lesson  easily  within  the  comprehension  of  young 
boys  at  the  end  of  their  First  Year.  Note:  (1)  the  proofs 
though  of  the  simplest  kind  are  enough  to  convince  young 
boys;  (2)  there  is  a  logical  grouping  of  the  different  kinds 
of  parallelograms;  (3)  the  gradual  extension  of  the  properties, 
as  the  variety  within  the  species  becomes  more  particularized, 
is  brought  out.  This  gradual  extension  of  properties  should 
always  be  borne  in  mind  in  the  teaching  of  geometry. 

Quadrilaterals  as  Parallelograms 

91.  A  quadrilateral    is    a   plane  rectilineal   figure 
with    four    sides.      (L.)     There    are    different    kinds    of 
quadrilaterals.     We  will  begin 

with  the  parallelogram.  /  /       /      / / 

92.  Draw    a    few    parallel  /  /        /       / / 

transversals  across  the  parallel         /  /        /        / / 

lines  of  your  notebook.     You  /  /       /       /           / 

see    a    number   of   four-sided  Fig.  91 

figures  with  their  opposite  sides 

parallel.    These  are  parallelograms.     (Gram  means  line.) 

How  can  we  define  a  parallelogram?  First  put  it  into 
its  class.  (§  73.) 

A  parallelogram  is  a  quadrilateral  .  .  . 


262 


CRAFTSMANSHIP  IN  MATHEMATICS 


What  special  property  distinguishes  a  parallelogram?     Its 
opposite  sides  are  parallel.    Hence  the  definition: 

A  parallelogram  is  a  quadrilateral  with  its  opposite 
sides  parallel.  (L.)  Now  let  us  discover  the  other  properties 
of  a  parallelogram. 

93.  Any  side  of  a  parallelogram  may  be  regarded  as  a 
transversal  across  two  parallel  lines.  Let  ABCD  be  a  parallelo- 
gram,    with    the     four     angles 
marked  as  shown. 


Fig.  93 


a  +  p  =  2  rt.  Zs, 

p  4.  y  =  2  rt.  Zs; 
/.   a  -f  (3  =  p  +  y. 

/.   a  -  y. 
Similarly,  p  =  8. 


(§84) 
(§84) 


Thus,  the  opposite  angles  of  a  parallelogram  are  equal. 


Fig.  93 


94.  Join  two  opposite  vertices  by  a  line.     Such  a  line  is 
called  a  diagonal.    This  diagonal  (AC)  is  a  new  transversal 

to  both  pairs  of  parallel  lines. 
Mark  the  two  pairs  of  equal 
alternate  angles,  al9  «2,  ft,  ft, 
and  the  pair  of  equal  opposite 
angles,  y1?  y2.  In  the  two  Z.s 
ABC,  ADC,  the  diagonal  forms 

a  side  belonging  to  both,  and  the  three  angles  of  the  one 
are  equal,  respectively,  to  the  three  angles  of  the  other. 
Hence  the  two  As  are  congruent. 

.'.  AB  =  CD    and    AD  =  BC. 

Thus,  the  opposite  sides  of  a  parallelogram  are 
equal.  (L.) 

95.  Draw   the   two  diagonals,    intersecting   at    E;    and 
consider  the  two  triangles  AEB,  CED.     In  these   As,  the 
three  /.s  of  the  one  are  equal  to  the  three  /.s  of  the  other 
(§  82),  and  AB  =  CD  (§  94).     Hence  the  As  are  congruent. 

/.  AE  =  CE    and    BE  =  DE. 


ELEMENTARY  GEOMETRY 


263 


Thus,  the  diagonals  of  a 
parallelogram  bisect  each 
other.  (L.)  (Of  course  we 
might  have  used  the  other  As 
AED  and  EEC.) 

96.  We  may  now  collect 
up  the  various  properties  of  a 
parallelogram: 


Fig.  94 


In  any  Parallelogram, 

(1)  The  opposite  sides  are  parallel,    (known  from  the 


(2)  The  opposite  angles  are  equal. 

(3)  The  opposite  sides  are  equal. 


definition) 
(proved) 
(proved) 


(4)  The  diagonals  bisect  each  other,   (proved) 

97.  Imagine  a  £17  m  to  be  jointed  at  the  four  angles,  and 
let  it  move  from  position  A  to  position  C.     It  remains  a 


Fig.  95 

ZZ7m  all  the  time,  and  therefore  keeps  all  its  properties.  But 
on  its  journey  to  C  it  passes  through  B,  where  the  angles 
are  right  angles.  A  parallelogram  with  right  angles  is  called 
a  rectangle.  A  RECTANGLE  is  a  right-angled  parallelo- 
gram. (L.)  Since  a  rectangle  is  a  ZI7m,  it  has  all  the 
properties  of  a  £I7m  (§  96);  and  it  has  certain  additional 
properties: 

98.  (i)  The  angles  of  a  rectangle  are  right  angles. 
(This  follows  from  the  definition.) 

99.  (ii)  Draw  the  two  diagonals  of 
the  rectangle  ABCD,  and  examine  the 
two   As  ABC  and  DCB  (which,  it  will 
be   seen,    partly    overlap    and    have   a 

common  base).     The  two  sides  AB,  BC  are  equal  to  the 


Fig.  96 


264  CRAFTSMANSHIP  IN  MATHEMATICS 

two  sides  DC,  CB  (§  96,  3);  and  the  included  angles  ABC 
and  DCB  are  equal,  both  being  right  angles.  Hence  the 
As  are  congruent.  Therefore  AC  is  equal  to  BD.  Thus, 
the  diagonals  of  a  rectangle  are  equal.  (L.) 

100.  Suppose  a  rectangle  gets  shorter  and  shorter  until 
its  length  and  breadth  are  equal.      It  remains  a  rectangle 


Fig.  97 

all  the  time  and  therefore  keeps  all  its  properties.  When 
the  length  exceeds  the  breadth,  as  in  A  and  B,  the  rectangle 
is  called  an  oblong;  when  its  length  and  breadth  are  equal, 
it  is  called  a  square. 

101.  An  OBLONG  is  a  rectangle  with  its  length 
exceeding  its  breadth.    (L.) 

102.  A  SQUARE  is  a  rectangle  with  all  four  sides 
equal.     (L.)     Since  a  square  is  a  rectangle,  it  has  all  the 
properties    of   a    rectangle  (§§  97-99);    and    it    has    certain 
additional   properties: 

103.  (i)  All  four  sides  of  a  square  are 
equal.     (This  follows  from  the  definition.) 

104.  (ii)    Draw    the    two   diagonals    of   the 
square  ABCD,  intersecting  in  E,  and  examine 

Fig. 98         the  two  As  EEC  and  DEC. 

BE  =  DE,  (§§  95,  97) 

BC  -  DC,  (§  103) 

EC  is  common  to  both  As; 
/.    ABEC  =  A  DEC; 
.'.    ZBEC  =  /.DEC; 

.*.  each  Z  =  J  st.  Z  or  1  rt.  Z. 

Similarly  we  may  show  that  each  of  the  other  two  Zs  at 
E  are  rt.  Zs.  Hence,  the  diagonals  of  a  square  bisect 
each  other  at  rt.  Zs.  (L.) 

105.  We  still  require  names  for  the  two  non-rectangular 
parallelograms.  The  non-rectangular  parallelogram  with  all 
four  sides  equal  is  called  a  rhombus  (fig.  99,  i).  The  non- 


ELEMENTARY  GEOMETRY 


265 


Fig.  99 


rectangular  parallelogram  with  only  its  opposite  sides  equal 
is  called  a  rhomboid  (ii).  The  rhomboid  is  the  parallelogram 

we  began  with  (§§   93-96).     It          

is  the  most   general  form  of    /     /  / / 

parallelogram.     If  we  made  all          i 

its  sides  equal,  or   if  we  made 

all  its  angles  right  angles,  we  should  make  it  a  particular 

kind  of  parallelogram. 

106.  Just  as  an  oblong  may  be  reduced  in  length  and 
made  a  square,  so  a  rhomboid  may  be  reduced  in  length 
and  made  a  rhombus. 

Just  as  a  square  has  all  the  properties  of  an  oblong  and 
certain  additional  properties,  so  a  rhombus  has  all  the  pro- 
perties of  a  rhomboid  and  certain  additional  properties. 

107.  All  four  sides  of  a  rhombus  are  equal.    (This 
follows  from  the  definition.) 

108.  The  diagonals  of  a  rhombus  are  the  perpen- 
dicular bisectors  of  each  other.      (L.)     This  property 
may  be  discovered  in  this  way.    The  word 

rhombus  really  means  a  spinning-top.     If 

we  stand  it  on  an  angle,  it  looks  something 

like  a  spinning-top.     We  see  at  once  that 

either  diagonal  is  an  axis  of  symmetry,  B 

and  D  being  corresponding   points  about 

the  axis  AC,  and  A  and  C  being  corresponding  points  about 

the   axis    BD.      Thus    each   diagonal    is    the    perpendicular 

bisector  of  the  other. 

But  this  also  applies  to  a  square.  What  is  the  difference 
between  a  rhombus  and  a  square? 


Fig.  101 


109.  If  we  lengthen  (or  shorten)  equally  the  two  halves 
of  one  of  the  diagonals,  AB,  of  a  square,  we  stretch  out  (or 


266  CRAFTSMANSHIP  IN  MATHEMATICS 

contract)  the  square  into  a  rhombus.     The  diagonals  of  a 
square  are  equal;   those  of  a  rhombus  are  unequal. 
A  rhombus  differs  from  a  square  in  these  ways: 

1.  Its  angles  are  not  right  angles. 

2.  Its  diagonals  are  not  equal. 

A  rhombus  resembles  a  square  in  these: 

1.  The  four  sides  are  equal. 

2.  Each  diagonal  is  the  perpendicular  bisector  of  the 

other. 

3.  Each  diagonal  is  an  axis  of  symmetry. 

110.  We  may  classify  the  four  kinds  of  parallelograms  in 
this  way: 

Parallelograms 


Rectangles  Non- rectangles 


Square  Oblong  Rhombus  Rhomboid 

(all  sides  eq.)      (only  opp.  sides  eq.)      (all  sides  eq.)      (only  opp.  sides  eq.) 

How  easy  it  is  to  make  up  definitions  from  this  scheme: 

111.  A  SQUARE  is  a  rectangular  parallelogram 
with  all  four  sides  equal  (§  102). 

112.  An  OBLONG  is  a  rectangular  parallelogram 
with  only  its  opposite  sides  equal  (§  101). 

113.  A  RHOMBUS  is  a  non -rectangular  parallelo- 
gram with  all  four  sides  equal.    (L.) 

114.  A  RHOMBOID  is  a  non -rectangular  parallelo- 
gram with  only  its  opposite  sides  equal.    (L.) 

115.  We  might  classify  parallelograms  according  to  their 
axes  of  symmetry.   A  square  has  four  axes  of  symmetry,  viz. 
two  diagonals  and  two  medians  (a  median  is  the  line  joining 
the  middle  points  of  opposite  sides);    an  oblong  has  two, 


ELEMENTARY  GEOMETRY  267 

viz.  the  two  medians;  a  rhombus  has  two,  viz.  the  two 
diagonals;  a  rhomboid  has  none.  But  in  definitions  we 
do  not  usually  refer  to  symmetry;  symmetry  is  useful  mainly 
for  discovering  properties. 

Remember  that  the  two  halves  of  a  figure  folded  on  an 
axis  of  symmetry  will  fit  together  exactly.  Remember,  too, 
that  a  figure  can  always  be  imagined  to  spin  on  an  axis  of 
symmetry. 

116.  More  about   definitions.      We    have   defined   a 
square  as  a  rectangle  with  all  four  sides  equal.   There- 
fore, 

(i)  Since  it  is  a  rectangle,  it  has  right  angles  and  is  a 
parallelogram  (§  97). 

(ii)  As  it  is  a  parallelogram,  its  opposite  sides  are 
parallel  (§  92).  Thus,  our  definition  of  a  square  tells 
us  three  things: 

1.  The  four  sides  are  equal, 

2.  The  four  angles  are  right  angles, 

3.  The  opposite  sides  are  parallel. 

But  the  definition  tells  us  nothing  at  all  about  the  diagonals. 
Properties  of  the  diagonals  must  be  discovered  either  by 
congruence  (§  104)  or  by  symmetry. 

117.  We  might  define  a  square  as  a  quadrilateral  with 
four  equal  sides  and  four  right  angles.    But  all  that  this  defini- 
tion tells  us  is  that: 

1.  The  four  sides  are  equal, 

2.  The  four  angles  are  right  angles. 

It  is  quite  a  good  definition,  but  it  does  not  tell  us  that  the 
square  is  a  parallelogram,  and  therefore  it  does  not  tell 
us  that  the  opposite  sides  are  parallel.  Hence  this  property 
is  one  we  should  have  to  find  out  (perhaps  by  congruence)  if 
we  used  the  new  definition.  Let  us  decide  not  to  use  it. 

118.  We  can  now  classify  the  properties  of  parallelo- 
grams: 


268 


CRAFTSMANSHIP  IN  MATHEMATICS 


Rhomboid. 

Oblong. 

Square. 

Rhombus. 

£3 

a 

D 

n 

1.  Opp.  sides  ||. 

X 

X 

X 

X 

2.  Opp.  sides  eq. 

X 

X 

X 

X 

3.  Opp.  Zs.  eq. 

X 

X 

X 

X 

4.   Diags.  bisect  each  other. 

X 

X 

X 

X 

5.  All  four  Zs.  rt.  Zs. 

X 

X 

6.  Diags.  eq. 

X 

X 

7.  All  four  sides,  eq. 

X 

X 

8.  Diags.  at  rt.  Zs. 

X 

X 

Observe  that  the  rhomboid  is  the  most  general  of  the  parallelo- 
grams, and  has  fewest  properties;  and  that  the  square  is  the 
most  special  of  the  parallelograms,  and  has  most  properties. 
No  other  parallelogram  has  all  the  properties  of  the  square. 

This  lesson  may  usefully  be  followed  up  by  the  considera- 
tion of  quadrilaterals  that  are  not  parallelograms. 


Proportion  and  Similarity 

A  knowledge  of  proportion  and  similarity  is  so  fruitful 
throughout  the  whole  range  of  the  study  of  geometry  that 
the  subject  should  be  introduced  at  an  early  stage,  though 
naturally  incommensurables  are  then  ignored  entirely.  We 
append  a  lesson  suitable  for  the  second  year  of  the  geometry 
course.  Note  the  little  device  for  constructing  a  triangle 
with  sides  simply  commensurable.  The  proofs  given  are 
rigorous  enough  at  this  early  stage.  The  important  thing  is 
to  provide  learners  with  a  serviceable  weapon — rough  and 
unpolished,  for  the  moment,  it  is  true;  but  that  is  of  no 
consequence. 

(The  nature  of  a  ratio,  of  cross-multiplication,  &c.,  has 
already  been  referred  to  in  the  chapters  on  arithmetic  and 
algebra,  but  the  three  subjects  should  be  brought  into  line 
when  a  principle  common  to  them  all  is  under  consideration.) 


ELEMENTARY  GEOMETRY  269 

119.  Take  a  piece  of  paper  ruled  in  J"  squares,  and  on 
it  draw  this  triangle:  the  base  AB  of  the   A  is  to  be  on  one 
of  the  ruled  horizontal  lines  4"  or  5"  down  the  paper,  and 
the   vertex   in  a   parallel    line    3" 

above,  i.e.  in  the  twelfth  parallel 

line    above.      Fix    the    point    A 

towards  the  left-hand  end  of  the 

line  selected  for  the  base,  and  with 

a  radius  of  3-6"  draw  a  circle  to 

cut  the  top  line  in  C.      With  C  Fig.  102 

as   centre,   and   with  a  radius  of 

4-5",  draw  a  circle  to  cut  the  base  line  in  B.     Join  AB  (its 

length  does  not  matter)  and  so  complete  the   A  ABC. 

Since  AC  =  3-6",  it  can  be  divided  into  twelve  equal 
parts  of  -3"  each,  and  each  division  will  fall  on  one  of  the 
ruled  horizontal  lines.  Since  CB  ^  -  4i",  it  can  be  divided 
into  twelve  parts  of  f "  each,  and  again  each  division  will  fall 
on  one  of  the  ruled  horizontal  lines.  But  the  ruled  lines  are 
all  parallel  to  each  other.  We  therefore  seem  to  have  the 
following  result: 

120.  If  the  two  sides  of  a  triangle  are  divided  into 
the  same  number  of  equal  parts,  and  the  corresponding 
points  of  division  in  the  two  sides  are  joined,  all  the 
joining  lines  are  parallel  to  the  base.   (L.) 

It  has  been  found  that  this  result  is  always  true,  no  matter 
how  it  is  tested.  But  the  real  proof  is  too  difficult  for  you 
to  understand  at  present.  The  following  particular  case  is 
often  useful: 

121.  If  two  sides  of  a  triangle  are  bisected,  the  line 
joining  the  points  of  division  is  parallel  to  the  base. 
(L.) 

122.  We  may  now  learn  that  if 
a  line  is  drawn  parallel  to  one  side 
of  a  A ,  it  cuts  the  other  side  pro- 
portionally.    Consider,  for  instance, 
the  fifth  parallel  DE,  from  the  top 

in  the  figure  to  §  119.     CD  is  -x\  of  Flg>  I03 


270  CRAFTSMANSHIP  IN  MATHEMATICS 

CA,  and  DA  is  -&  of  CA;  CE  is  -&  of  CB,  and  EB  is  -/2-  of 
CB. 

CD       5          ,     CE       5 

•  —  nnn       — 

"   DA~7  EB~7' 

CD  ^  CE 

"   DA    "  EB' 

So  with  any  other  parallel.  Or  a  part  of  a  side  may  be  compared 
with  the  whole.    For  instance, 

CD  ^  CE 

CA  ~  CB' 

for  each  is  equal  to  the  fraction  -L!V 

We  can  imagine  the  A  CDE  to  be  a  small  A  fitting  over 
the  top  of  the  larger  A  CAB  (fig.  104,  i),  and  CD  being  made 
to  slide  down  CA  so  that  the  small  A  CDE  occupies  the 


\L _\r 


Fig.  104 


position  C'D'E',  D  taking  the  place  of  A  (fig.  104,  ii).  Just 
as  the  three  /.s  of  the  small  A  are  respectively  equal  to  the 
three  Ls  of  the  large  A  in  fig.  104,  i  (see  §  78),  so  they 
must  be  in  the  second,  since  corresponding  angles  are  equal. 
Hence  C'E'  is  ||  CB,  and  therefore  C'E'  cuts  the  two  sides 
D'C  and  D'B  proportionally;  and  just  as  C'D'  is  -{\  of  CD', 
so  D'E'  must  be  ^  of  D'B, 

D'C'       D'E'  D'C'       D'E' 

or =     — .     or     = . 

C'C        E'B'  D'C        D'B 

Similarly  by  making  the  little  A  slide  down  to  the  other 
corner  (fig.  104,  i  to  iii),  so  that  E  takes  the  place  of  B,  C"D" 
is  ||  to  CA,  and  therefore 

E//C"      E"D"  E"C"      E"D" 

-—      .  Q|«  — • 

C"C         D"A'  E"C         E"A' 

Hence: 


ELEMENTARY  GEOMETRY  271 

123.  If  in  a  triangle  a  line  is  drawn  parallel  to 
any  side,  it  cuts  the  other  sides  proportionally.    (L.) 
(This  is  always  true,  but  the  real  proof  is  too  difficult  for 
you  to  understand  at  present.) 

124.  We  might  detach  the  small  ACDE  from  the  large 
one  CAB,  and  place  the  two  side  by  side.    They  look  alike 
They  are  alike.     They  are  similar.  c 

Although  the   sides   of  the   two    As 
differ  so  much  in  length,  the  three 

Z.s  of  the  one  are  respectively  equal  to     A/ \B 

the  three   /.s  of  the  other.     (Why?)  blg.  I05 

In  other  words,  the  two  similar  As  are 

equiangular.  And  we  know  already  that  the  corresponding 
sides  are  proportional.  This  we  should  expect  in  similar 
figures  of  any  kind.  In  a  photograph  of  yourself,  for 
instance,  you  would  expect  the  "  proportions  "  of  your  body 
to  be  accurately  preserved.  If  the  ratio  of  the  lengths  of 
your  outstretched  forearm  and  upper  arm  is  f ,  you  would 
expect  that  ratio  to  be  preserved  in  the  photograph  (or  the 
photographer  would  probably  hear  about  it!). 

125.  If,  then,  ABC  and  DEF  are  two  similar    As,  the 
corresponding   sides   are   proportional.      But   note   that   we 
may   express    the    ratios    in   two 

different  ways:  (1)  two  sides  of 
one  A  as  a  ratio  equal  to  the 
ratio  of  the  corr.  two  sides  of  the 
other  A ;  (2)  one  side  of  one  A  plfg.  I06 

and  the  corr.  side  of  the  other 

A  as  a  ratio  equal  to  the  ratio  of  any  second  side  of  the 
first  A  and  the  corr.  side  of  the  second  A .  Consider,  for 
instance,  the  two  sides  AB,  BC  in  the  A  ABC,  and  the  two 
corr.  sides  DE,  EF  in  the  A  DEF.  We  may  say, 

AB       DE  AB       BC 

BC'EF'    °r    DE=EF- 

The  two  proportional  statements  are  really  the  same  thing 


272  CRAFTSMANSHIP  IN  MATHEMATICS 

since   we   obtain  the   same   product   from  the   cross-multi- 
plication of  either: 

AB  .  EF  =  BC  .  DE.     (The  full  stop  is  used  instead  of  X  .) 

It  is  often  an  advantage  to  interchange  one  form  for  the 
other;    really  we  interchange  the  second  term  of  the  first 
ratio  and  the  first  term  of  the  second. 
We  have  learnt  that: 

126.  SIMILAR  TRIANGLES  are  equiangular,  and 
their  corresponding  sides  are  proportional.    (L.) 

127.  When  expressing  ratios  between  two  sides  of  each 
of  two   similar    As,    be    careful    to   select   corresponding 


sides,  i.e.  sides  taken  in  the  same  order  round  corre- 
sponding angles.  In  these  two  pairs  of  As,  the  corre- 
sponding Zs  are  marked  with  the  same  Greek  letters.  From 
the  first  pair  we  may  equate  ratios  thus,  six  equations  in  all: 

AB  =  DE.  AB  _  DE.  AC  _  OF.  AB  _  BC.  AB  _  AC.  AC  _  BC 
BC  EF;  AC  DF;  CB  FEJ  DE  EF;  DE  DF;  DF  EF* 

From  the  second  pair,  we  may  do  exactly  the  same  thing: 

=Q5.  MP^QS.  MN^NP.  M^MP.  MP__PN 
QS;PN    SR;  QR    RS;  QR    QS;QS    SR* 

Yet  there  appears  to  be  a  difference.  That  is  because  in 
the  second  pair  the  As  are  right-  and  left-handed.  If  you 
have  any  doubt,  turn  one  of  the  pair  over,  through  180°, 
as  you  would  turn  over  a  page  of  a  book.  Then  the  pair 
will  look  alike.  But  if  you  mark  the  corresponding  angles 
correctly,  you  ought  to  have  no  difficulty. 


ELEMENTARY  GEOMETRY 


273 


The  sides  of  the  triangles  may  usefully  be  named  by 
means  of  single  small  letters;  then  the  writing  of  the  ratios 

is  simplified;  e.g.  -  instead  of  — ..     The  small  letter  selected 
a  BC 

for  a  side  is  always  the  same  as  the  capital  letter  naming  the 
opposite  angle. 

128.  Sometimes  each  of  a  pair  of  similar  As  is  similarly 
divided  by  a  perpendicular  from  a  vertex  to  the  opposite 
side.  The  resulting  pair  of  A 

As  in  the  one  case  are  evi- 
dently similar,  respectively, 
to  the  resulting  pair  in  the 

other     case,    for     they    are 

.  ~~  u  ~  " 

equiangular,   i.e.    As    ABG  iv]g.  10s 

and   DEH   are   similar,  and 

As  ACG  and   DFH   arc  similar  (check,  by  sum  of  angles). 
We  may  reason  in  this  way: 


In  the  As  ABG,  DEH,  since 


AG_  =  AB 
DH       DE 

4,         4, 

AB 


and  in  the  As  ABC,  DEF,  since  —  = 

I 
.    AG  _  BC 

"  DH  EF* 


An  exchange  of  ratios  may  often  usefully  be  made  in  this 
way.  From  this  particular  exchange  we  learn  that  the  altitudes 
AG,  DH  are  proportional  to  the  bases  BC,  EF.  Hence: 

129.  When  similar  triangles  are  divided  by  perpen- 
diculars drawn  from  corresponding  vertices  to  opposite 
sides,  an  exchange  of  ratios   may  often  be  usefully 
made.     (L.) 

130.  In  similar  triangles,  the  altitudes  are  pro- 
portional to  the  bases.    (L.) 

131.  We  know  that  when  two  ratios  are  equated  to  form 
a  proportion,  they  may   be  cleared   of  fractions   by  cross- 

(E291)  19 


274 


CRAFTSMANSHIP  IN  MATHEMATICS 


multiplication.  For  instance,  in  the  two  similar  As  ABC, 
DEF,  we  know  that 

—  =- ;  (an  equation  consisting  of  2  ratios) 

BC       EF 

/.  AB  .  EF  =  DE  .  BC.      (an  equation  consisting  of  2  products) 

What  does  this  mean?  AB,  BC,  DE,  EF,  all  represent  lines 
of  a  particular  length;  a  length  multiplied  by  a  length  gives 
an  area.  Thus  each  of  the  two  products  AB  .  EF  and  DE  .  BC 
represents  a  rectangle. 


c    e 


6-25 


Fig.  109 


Note  that  we  begin  with  ratios,  i.e.  with  quotients, 
representing  a  length  divided  by  a  length.  After  cross- 
multiplying,  we  have  products  representing  areas,  or  a 
length  multiplied  by  a  length.  (The  measured  lengths  are 
shown  to  scale.  Check  the  numerical  ratios  and  the  products. 

6-25           3'75 
For  instance,  are and equal?    and  are  10  X  3-75 

and  6X6- 25  equal?)  10  6 

132.  By  cross -multiplication,  two  equated  ratios 
of  lengths  give  two  equated  rectangular  areas. 

("  Equated  "  means  expressed  as  an  equation.) 

133.  The  last  result  is  useful  in  all 
sorts  of  ways.  For  instance,  AB  and 
CD  are  two  chords  of  a  circle,  inter- 
secting at  O.  Join  AC  and  DB,  and 
we  have  two  similar  As,  the  As  being 
equiangular  (angles  in  the  same  segment; 
see  §  126).  Taking  ratios  (see  §  127) 
we  have, 


Fig.  no 


OA 

oc: 

rect.  OA .  OB 


op 

OB  ; 

rect.  OC  .  OD. 


ELEMENTARY  GEOMETRY 


275 


Hence,  if  two  chords  intersect  in  a  circle,  the  rectangle 
contained  by  the  two  segments  of  the  one  is  equal  to 
the  rectangle  contained  by  the  two  segments  of  the 
other.  (L.)  (The  term  "  segment  "  here  applies  to  the 
parts  of  the  chords.) 


Fig.  in 

The  proportional  division  of  lines  and  the  construction  of 
similar  figures  should  follow  at  a  slightly  later  stage.  Be  sure 
that  the  boys  master  the  principle  exemplified  in  these  three 
figures:  for  the  construction  of  the  similar  pentagon  the 
position  of. the  point  O  is  quite  immaterial. 

The  centre  of  similarity  problems  are  readily  followed  by 
those  on  centre  of  similitude.  Insist  on  the  point  that  any 
two  circles  may  be  regarded  as  similar  figures,  since,  like 
rectilineal  similar  figures,  they  may  be  looked  upon  as  the 
same  figure  drawn  to  different  scales. 


Circles  and  Polygons 

The  ordinary  properties  of  the  circle  give  little  trouble — 
angles  in  a  segment,  the  cyclic  quadrilateral,  tangents, 
alternate  segment  property,  circles  in  contact,  and  inter- 
secting circles.  Do  not  forget  to  group  properties  around  a 


276 


CRAFTSMANSHIP  IN  MATHEMATICS 


common  principle;  e.g.  (i)  the  tangent  to  a  circle,  (ii)  the 
external  common  tangent  to  two  circles,  (iii)  the  transverse 
common  tangent  to  two  circles,  should  be  taken  in  that  order, 
and  be  made  to  follow  on  the  key  proposition  that  the  angle 
in  a  semicircle  is  a  right  angle.  All  these  propositions  on  the 
circle  being  quite  simple,  formal  proofs  should  now  be 
consistently  exacted. 

Regular  polygons,  too,  need  give  little  trouble.  Their 
angle  properties  are  interesting,  easy  to  understand,  and 
always  appeal  to  a  boy.  The  pentagon  excepted  (see  the 
next  section),  they  are  not  much 
wanted.  The  hexagon  and  octagon 
involve  the  simplest  geometry,  easy 
work  for  beginners.  The  decagon  is 
easily  constructed  from  the  pentagon, 
and  the  dodecagon  from  the  hexagon. 
The  heptagon  and  nonagon  are  hardly 
ever  used;  the  latter  is  easily  constructed 
from  its  angle  properties;  the  former 
is  riot,  inasmuch  as  its  angle  properties 
involve  fractions  of  a  degree  and  hence 
some  sort  of  approximation  method  is 
required  for  its  construction.  The 

best  is  probably  the  following,  especially  as  it  is  common  to 
all  polygons. 

On  one  side  of  a  straight  line  draw  a  semicircle  and  on 
the  other  side  an  equilateral  triangle.  If  the  line  be  divided 
into  x  equal  parts,  and  lines  be  drawn  from  the  apex  of  the 
triangle  through  the  points  of  division,  to  meet  the  semi- 
circle, the  semicircle  is  divided  into  the  same  number  of 
parts  as  the  line.  This  is  not  susceptible  of  proof,  simply 
because  it  is  not  mathematically  true,  but  the  approximation 
is  so  near  that  the  most  careful  measurement  usually  fails 
to  detect  an  error.  Architects  generally  use  it.  Evidently 
by  drawing  radii  from  the  points  of  division  of  the  semicircle, 
we  divide  180°  into  x  equal  parts. 

The  conversion  of  polygons  (regular  and  irregular)  into 


ELEMENTARY  GEOMETRY  277 

triangles,  triangles  into  rectangles,  and  rectangles  into 
squares,  which  is  often  wanted,  is  simple  straightforward 
work,  though  some  little  practice  in  manipulating  the  figures 
is  necessary.  To  the  beginner,  a  polygon  with  one  or  more 
re-entrant  angles  is  puzzling. 

Golden  Section  and  the  Pentagon 

We  append  the  following  lesson  as  an  example  of  linking 
up  different  Euclidean  propositions  (II,  11;  IV,  10,  11)  and 
of  utilizing  algebra  in  solving  geometrical  problems. 

134.  To  divide  a  line  into  two  parts  so  that  the 
rectangle  contained  by  the  whole  and  one  part  is  equal 
to  the  square  on  the  other  part. 

This  is  sometimes  stated: 

To  divide  a  line  in  medial  section. 
or,  To  divide  a  line  in  extreme  and  mean  ratio. 
or,  To  divide  a  line  in  golden  section. 

The  problem  is  very  easy  to  do  and  to  understand  if  we 
can  solve  easy  quadratic  equations.  It  is  the  kind  of  problem 
in  which  algebra  can  help  us 

much,  A,  x  L(2-z) >B 

Let    AB    be    the    line    to  Fig  II3 

be  divided,  and  let  it  be,  say, 

2"  long.     Suppose  the  point  of  division  is  P. 

Let  AP  be  x  inches  long;  then  PB  =  (2  —  x)  inches  long. 
The  line  has  to  be  divided  so  that  AB  .  BP  =  AP2, 

i.e.  2(2  —  x)  =  x2. 
We  must  now  solve  this  equation,  and  find  the  value  of  x 

X*  =  2(2  -  *); 
.'.   x2  +  2x  =  4. 
/.  x2  +  2x  +  I  =  5. 

.*.  x  +  I  =  +  V5.     (We  may  neglect  the  minus  sign.) 
/.  X  =  V5  -  1, 


278 


CRAFTSMANSHIP  IN  MATHEMATICS 


i.e.  AP  =  (\/5  — •  1)  inches.  Can  we  measure  off  this  length 
and  so  find  P?  Yes,  by  the  theorem  of  Pythagoras.  We  do 
it  in  this  way: 

Erect  a  JL  BC  at  B,  1"  long,  and  join  CA. 

AC2  -  (AB2  +  BC2)  =  (22  +  I2)  =  5; 
/.  AC  =  V5, 
i.e.  AC  is  V5  inches  long  (fig.  113a,  i). 


Fig.  1130 

But  we  require  a  line  (\/5  —  1)  inches  long. 
Since  CB  —  I",  with  centre  C  and  radius  CB,  cut  CA  in 
D  (fig.  1130,  ii);    CD  =  I". 

Thus  AD  -  (V5  -  1)  inches. 

But  we  require  a  part  of  AB  equal  to  (\/5  —  1)  inches. 
Hence,   with   centre  A   and   radius  AD,   cut  AB  in   P; 
AP  =  (<v/5  —  1)  inches. 

Thus  P  is  the  point  required. 

The  length   of  PB   is  evidently   2  —  (<v/5  —  1)   in.,   i.e. 
(3  —  V5)  inches. 

If  we  have  done  "  surds  "  in  algebra,  we  can  show 
that  the  result  is  correct:  AB  .  BP  has  to  be  equal  to 
AP2.  Now  AB  .  BP  -  2(3  -  ^5)  =  6  -  2  V5;  and  AP2  - 
(V5  -  I)2  =  6  -  2-v/5,  as  before. 

Here  is  Euclid's  figure. — He  does  not 
cut  off  a  piece  from  CA;  he  makes  CD 
equal  to  CA,  so  that  CD  =  <v/5,  and  BD 
==  \/5  —  !•  Then  he  makes  BP  equal  to 
BD  =  \/5  —  1,  so  that  P  is  found  as  before, 
except  that  PB  is  now  the  longer  instead 
Fig.  ii4  of  the  shorter  section.  The  shaded  parts 

of   the  figure  show  the  rectangle  AB  .  AP 
2(3  —  V5);  and  the  square  on  PB,  (^/5  —  I)2. 


ELEMENTARY  GEOMETRY 


279 


Now  examine  a  regular  pentagon  and  its  5  contained  dia- 
gonals. Give  the  boys  a  few  hints  (such  as  the  following) 
and  then  leave  them  to  construct  the  pentagon  themselves. 


(1)  Angles. — 04  =  a2  =  a3;     hence    it    is    clear   that   the 
15  angles  at  the  5  vertices  of  the  pen- 
tagon are  all  equal,  and  that  each  = 

36°. 

(2)  Lines. — Each  diagonal  is  divided 
by  2  others   into   3   parts.      Is  there 
any  relation  between  the  parts?    e.g. 
does  CF  bear  any  relation  to  FA  or  to 
the  side  CD? 

Draw  the   triangle  ACD   and   the 
line  FD  separately,  and  write  in  all 

the  angles.  Evidently  AFD  and  DFC  are  isosceles  triangles. 
.• .  AF  =  FD,  CD  =  FD,  / .  AF  =  CD.  Hence  if  we  put 
a  circle  round  the  triangle  AFD,  CD  is  a  tangent  (relation 


Fig.  115 


Fig.  116 


=  ?);  also  CA  is  a  secant;  / .  CF  .  CA  =  CD2;  / .  CF  .  CA 
=  FA2,  i.e.  CA  is  divided  at  F  in  golden  section. 

To  construct  a  pentagon,  therefore,  we  begin  by  drawing 
any  line  AC,  and  dividing  it  in  golden  section  in  F.  With 
A  as  centre  and  AC  as  radius,  we  draw  a  circle  (not  shown) 
and  draw  in  it  the  chord  CD  equal  to  FA,  and  then  join 
AD.  This  gives  us  the  triangle  ACD,  round  which  we 
circumscribe  a  circle  and  so  obtain  part  of  fig.  115;  to  obtain 
the  points  B  and  E  we  bisect  the  angles  ADC,  ACD. 


280  CRAFTSMANSHIP  IN  MATHEMATICS 

Teach  the  boys  one  or  two  special  ways  of  drawing  the 
pentagon;  e.g.  let  them  tie  into  a  simple  knot  a  strip  of  paper 


of  uniform  width.    It  is  a  useful  exercise  to  make  them  prove 
that  the  figure  produced  really  is  a  pentagon. 

The  Principle  of  Continuity 

This  is  an  ambiguous  term,  for  in  each  of  several  branches 
of  knowledge  it  is  given  a  special  significance.  Even  in  the 
single  subject  mathematics,  it  is  used  in  different  senses. 
One  standard  textbook  of  geometry  states:  "  The  principle 
of  continuity,  the  vital  principle  of  modern  geometry,  asserts 
that  if  from  the  nature  of  a  particular  problem  we  should 
expect  a  certain  number  of  solutions,  and  if  in  any  particular 
case  we  find  this  number  of  solutions,  then  there  will  be 
the  same  number  of  solutions  in  all  cases,  although  some  of 
the  solutions  may  be  imaginary.  For  instance,  a  straight 
line  can  be  drawn  to  cut  a  circle  in  two  points;  hence  we 
state  that  every  straight  line  will  cut  a  circle  in  two  points, 
although  these  may  be  imaginary  or  may  coincide.  Similarly 
we  may  say  that  two  tangents  may  be  drawn  from  any  point 
to  a  circle,  but  they  may  be  imaginary  or  coincident/'  * 

But  in  geometry  the  term  "  continuity  "  has  come  to  be 
used  more  loosely  than  that.  It  is  used  to  indicate  generality, 
a  generalizing  of  some  fundamental  principle,  or  the  grouping 
of  a  number  of  allied  instances  around  some  central  principle. 
We  give  a  few  instances  of  different  kinds,  from  which  the 

*  Lachlan,  Modern  Pure  Geometry. 


ELEMENTARY  GEOMETRY 


281 


reader  will  see  more  clearly  what  is  meant.  As  regards  the 
teaching  of  geometry,  the  principle  is  one  of  the  very  greatest 
importance. 

1.  The  particularizing  of  a  general  figure  and  the  extension 
of  properties.     We  have  already  given  an  instance  of  this  in 
the  lesson  on  parallelograms. 

2.  Varying  the  figure  to  include  different  cases.      These 
three  figures  tell  their  own  story.    If  the  parts  of  such  figures 


Fig.  118 


are  similarly  named,  as  a  rule  exactly  the  same  words  apply 
in  all  cases  to  the  proof.  What  difference  there  is  is  generally 
a  difference  of  mere  sign. 

3.  Generalizing  a  term  to  include  its  natural  extensions, 


Fig.  119 


e.g.  a  chord  as  a  secant,  and  a  secant  as  a  tangent.  From  the 
case  of  intersection  O  inside  the  circle,  we  pass  to  the  case  of 
intersection  outside  the  circle,  and  then  from  the  two  secants  to 
a  secant  and  a  tangent.  The  three  cases  may  first  be  separately 


282 


CRAFTSMANSHIP  IN  MATHEMATICS 


taken  and  then  generalized.  If  the  lettering  is  consistent, 
the  arguments  are  identical,  though  for  the  tangent-secant 
case  we  should  generally  argue  rather  differently.  In  all 
three  cases  we  have  two  similar  triangles,  OAC  and  OBD, 
and  OA/OC  =  OD/OB. 

Another  general  chord-secant-tangent  property  is  seen  in 
the  following  four  figures,  showing  the  measure  of  an  angle 


Fig.  1 20 

inscribed  in  a  circle  by  reference  to  the  intercepted  arcs; 
again  the  argument  may  be  made  perfectly  general. 

4.  The  extensions  of  Pythagoras  form  another  series.  It 
is  the  general  custom  nowadays  to  give  the  boys  Pythagoras 
towards  the  end  of  their  first  year,  to  serve  as  a  useful  working 
tool;  to  give  a  formal  proof  during  the  second  year,  and  to 
take  the  extensions  (Euclid  II,  12,  13)  a  few  months  later 
still;  most  boys  are  then  familiar  with  the  results  in  the 
following  form. 


AB2  =  AC2  -1-  CB2  exactly. 


AB2  = 
AC8  +  CB2  +  2BC.CP 


AC8  +  CB«  -  2BC.CP 


where  CP  is  the 
projection  of 
CA  on  BC. 
(L.) 


ELEMENTARY  GEOMETRY 


283 


But  figures  to  illustrate  the  extensions  are  less  often  pro- 
vided.    Here  is  a  suggestion: 


Fig.  121 


Fig.  (i)  illustrates  Pythagoras,  In  (ii)  compare  the  squares 
on  the  new  sides  BY,  YA  with  the  squares  on  the  old  sides 
BC,  CA.  In  (iii)  compare  the  squares  on  the  new  sides 
BZ,  ZA  with  the  squares  on  the  old  sides  BC,  CA.  The 
dissections  are  interesting,  though  they  tend  to  puzzle  slower 
boys. 

5.  Summing  the  exterior  angles  of  a  polygon:  "  walking 
the  polygon  ".  First  consider  an  ordinary  convex  polygon. 
Mark  in  the  angles  systematically:  "  always  turn  to  the 


284  CRAFTSMANSHIP  IN  MATHEMATICS 

left  ";    the  angle  to  be   worked    is  that  between  the  old 
direction  and  the  new. 


SUM  op  Ls.  •  2ir 


Fig.  1220 

Secondly,  a  polygon  with  one  re-entrant  angle: 


SuMOfU-4ir 


Fig.  1226 

Thirdly,  a  cross  polygon;   also  with  one  re-entrant  angle: 

c 


D       SUM  op  Ls 


Fig.  I22C 

The  point  in  this  example  is  to  see  how  exactly  the  same 
principle  is  followed  out:  always  turn  to  the  left,  always 
measure  the  exterior  angle  between  the  old  direction  (pro- 
duced) and  the  new.  The  result  must  always  be  a  multiple 

Of  27T. 

6.  Euclid,  Book  IL     Given  an  algebraic  basis,  suitable 


ELEMENTARY  GEOMETRY  285 

figures,  and  a  rational  grouping,  props.  4  and  7,  5  and  6, 
9  and  10,  can  be  taught  in  a  single  lesson.  Never  make  the 
boys  go  through  the  Euclidean  jargon;  life  is  not  long  enough. 
7.  The  Sections  of  a  Cone  (for  more  advanced  pupils). 
Let  a  plane  perpendicular  to  the  plane  of  the  paper  rotate 
round  the  point  P,  first  cutting  the  cone  ABC  parallel  to 
the  base;  then  obliquely  to  cut  the  slant  surface;  then  more 
obliquely,  parallel  to  AC  and  cutting  the  base;  then  per- 
pendicularly to  the  base  and  cutting  the  base.  Since  the 
motion  may  be  regarded  as  continuous,  we  should  expect 
no  sudden  changes  in  the  properties  of  the  curves  made  by 
the  rotating  plane  as  it  cuts  the  cone.  Why 
should  there  be?  The  boys'  knowledge  of  * 

geometry  ought  by  this  time  to  make  them      P     /  \ 
revolt  against   the   idea  of  any  fundamental 
differences  in  the  properties  of  the  curves. 
The  curves  may  all  be  described  as  conies 
possessing    certain  common  properties.     In 


\ 


particular  positions  the  curves  have  certain  Fig.  123 

additional  and  special  properties,  but  the 
common  properties  will  remain.  Let  the  boys  understand 
that  for  convenience  we  study  the  curves  separately  first, 
and  collectively  later.  But  make  them  see  at  the  outset 
that  the  circle  is  just  a  particular  case  of  an  ellipse,  just 
as  the  ellipse  is  a  case  of  the  more  general  conic.  The 
elliptic  orbit  of  the  earth,  for  example,  is  so  very  nearly  a 
circle  that  a  correct  figure  drawn  on  paper  is  virtually  in- 
distinguishable from  a  circle.  Astronomical  figures  are 
often  purposely  exaggerated. 

8.  The  Polyhedra  (see  Chapter  XXXVIII).  These  form 
an  even  better  illustration  of  the  principle  of  continuity  than 
those  already  cited. 

The  principle  applies,  in  fact,  to  the  whole  range  of  geo- 
metry. To  deal  with  a  proposition  as  an  unrelated  unit  is, 
generally  speaking,  to  offend  almost  every  canon  of  geometrical 
teaching. 


286 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  Principle  of  Duality 

This  is  best  exemplified  by  a  few  well-known  pairs  of 
theorems: 


1.  If   the    sides   of    a    triangle 
are  equal,  the  opposite  angles  are 
equal. 

2.  If  two   triangles   have   two 
sides     and     the     included     angle 
respectively  equal,   the   triangles 
are  congruent. 

3.  If    a    quadrilateral    be    in- 
scribed in   a   circle,   the   sum   of 
one    pair    of   opposite    angles    is 
equal  to  the  sum  of  the  other  pair. 

4.  If  a  hexagon  be  inscribed  in 
a  circle,   the  three  points  of  in- 
tersection   of    pairs    of    opposite 
sides  are  collinear. 


1.  If  two  angles  of  a  triangle 
are  equal,  the  opposite  sides  are 
equal. 

2.  If  two   triangles   have   two 
angles    and     the     included     side 
respectively  equal,   the   triangles 
are  congruent. 

3.  If  a  quadrilateral  be  circum- 
scribed about  a  circle,  the  sum  of 
one  pair  of  opposite  sides  is  equal 
to  the  sum  of  the  other  pair. 

4.  If    a    hexagon    be    circum- 
scribed about  a  circle,  the  three 
diagonal  lines  connecting  opposite 
angles  are  concurrent. 


Such  pairs  of  propositions  are  said  be  dual  or  reciprocal. 

There  is,  in  short,  a  remarkable  analogy  between  de- 
scriptive propositions  concerning  figures  regarded  as  as- 
semblages of  points  and  those  concerning  corresponding 
figures  regarded  as  assemblages  of  straight  lines.  Any  two 
figures  of  which  the  points  of  one  correspond  to  the  lines 
of  the  other  are  said  to  be  reciprocal  figures.  When  a  pro- 
position has  been  proved  for  any  figure,  a  corresponding 
proposition  for  the  reciprocal  figure  may  be  enunciated  by 
merely  interchanging  the  terms  point  and  line;  locus  and 
envelope*,  point  of  intersection  of  two  lines  and  line  of  inter- 
section through  two  points]  &c.  The  truth  of  the  reciprocal  or 
dual  proposition  may  usually  be  inferred  from  what  is  called 
"  the  principle  of  duality  ". 

The  teacher  should  always  be  on  the  look-out  for  examples 
of  this  principle  which  gives  boys  so  much  insight  into  geo- 
metry. Numerous  examples  of  concurrency  and  collinearity 
will  occur  to  him  at  once.  The  principle  is  especially  useful 


ELEMENTARY  GEOMETRY  287 

in  the  treatment  of  more  advanced  work,  for  instance  in  the 
theory  of  perspective  and  in  the  theory  of  the  complete 
quadrilateral  (tetrastigms  and  tetragrams). 


CHAPTER  XXI 

Solid  Geometry 

Preliminary  Work 

First  notions  of  solid  geometry  will  have  been  given  in 
the  Preparatory  School.  Even  in  the  Kindergarten  School 
the  children  are  made  acquainted  with  the  shapes  of  common 
geometrical  figures  and  solids.  Lower  Form  arithmetic  is 
closely  linked  up  with  practical  mensuration,  and  quite  young 
boys  are  made  familiar  with  the  methods  of  measuring  up 
rectangular  surfaces  and  solids.  The  practical  mensuration 
associated  with  early  measurements  in  physics  forms  another 
introduction  to  solid  geometry.  First  notions  of  projection 
are  given  in  early  geography  lessons;  very  young  boys  soon 
acquire  facility  in  building  up  vertical  cross-sections  from 
contoured  ordnance  maps,  and  when  projection  is  first  formally 
taken  up  in  the  mathematical  lessons,  say  in  the  Pythagoras 
extensions  or  in  early  trigonometry,  the  main  idea  is  already 
familiar.  All  the  way  up  the  school,  three-dimensional 
geometry  in  some  form  should  be  made  to  serve  as  a  hand- 
maid to  the  plane  geometry.  Indeed,  first  notions  of  the 
geometry  of  the  sphere  are  required  at  a  very  early  stage  in 
the  teaching  of  geography,  and  if  these  notions  are  to  be 
properly  implanted  the  mathematical  Staff  should  make 
themselves  responsible,  for  not  all  geography  teachers  are 
mathematicians. 

Only  a  minority  of  boys  acquire  readiness  in  reading 
geometrical  figures  of  three  dimensions.  With  the  majority, 


288  CRAFTSMANSHIP  IN  MATHEMATICS 

the  training  of  the  geometrical  imagination  is  a  slow  business. 
For  the  clear  visualization  of  the  correct  spatial  relations  in 
an  elaborate  three-dimensional  figure,  or  for  that  matter  even 
in  a  simple  one,  models  of  some  kind  are,  in  the  earlier  stages, 
essential. 

Supplies  of  useful  little  wooden  models  of  the  geometrical 
solids  are  often  found  in  the  physics  laboratory,  though  why 
physics  teachers  so  frequently  relieve  their  mathematical 
colleagues  of  this  particular  work  I  have  never  been  able  to 
discover.  If  models  in  wood  are  not  available,  models  may 
be  readily  cut  from  good  yellow  bar  soap;  the  material  is 
cleaner  to  handle  than  raw  potato  or  clay  or  plasticine.  By 
means  of  a  roughly-cut  model,  the  correct  shape  of  a  trans- 
verse section  of  a  geometrical  solid  can  be  realized  at  once. 
Personally  I  prefer  models  made  from  "  nets  "  of  cartridge 
paper  or  thin  cardboard;  these  are  easy  to  make  and  are 
permanent,  but  the  making  consumes  a  good  deal  of  time. 

Useful  skeleton  models  are  readily  made  from  pieces  of 
long  knitting  needles,  sharpened  at  each  end  and  thrust  into 
small  connecting  corks.  Two  slabs  of  cork  to  represent 
the  Horizontal  and  Vertical  Planes,  tacked  to  a  pair  of  hinged 
boards,  and  a  few  pointed  knitting  needles,  make  excellent 
provision  for  the  first  lessons  on  orthographic  projection. 

The  natural  sections  of  an  orange,  or  the  cut  sections  of 
a  well-shaped  apple,  are  useful  when  teaching  the  geometry 
of  the  sphere. 

The  small  varnished  wooden  models  of  the  cylinder, 
sphere,  and  cone,  of  the  same  diameter  and  height,  are  useful 
for  showing,  by  displacement  of  water  in  a  measuring  jar, 
that  the  volumes  are  3:2:1. 

A  slated  sphere,  mounted,  should  be  part  of  the  equipment 
of  all  mathematical  teachers. 

Even  such  a  simple  device  as  two  intersecting  sheets  of 
paper,  each  sheet  being  slit  half-way  across,  to  show  the 
intersection  of  two  planes  at  any  angle,  is  often  useful. 

But  of  course  all  these  props  should  gradually  be  with- 
drawn, and  the  eye  made  to  depend  on  two-dimensional 


SOLID  GEOMETRY  289 

drawings.  Still,  it  is  always  an  advantage,  even  for  the  trained 
mathematician,  to  put  a  few  shading  lines  into  such  drawings. 
They  help  the  eye  greatly. 

Stereographic  photographs,  or  even  hand-made  stereo- 
grams,  are  also  a  great  aid  in  teaching  solid  geometry.  These 
are  easily  provided,  and  stereoscopes  are  cheap.  Mr.  E.  M. 
Langley  used  them  with  great  effect  as  far  back  as  the  nineties. 

Do  not  forget  that  even  for  plane  geometry  models  may 
be  useful.  The  pantograph  is  particularly  useful  when  teaching 
similarity  (see  Carson  and  Smith's  Geometry).  When  teaching 
loci,  encourage  the  boys  to  make  wooden  or  cardboard 
"  linkages  "  to  represent  engineering  motions  and  astro- 
nomical movements.  The  loci  are  then  given  a  reality. 

The  boys  should  also  be  encouraged  to  make  "  nets  " 
of  the  commoner  geometrical  solids,  in  cartridge  paper  or 
cardboard.  Boys  of  11  or  12  learn  to  make  these  readily, 
and  at  that  age  time  can  be  spared.  I  have  known  boys  of 
10  make  almost  perfect  paper  models  of  the  five  regular 
polyhedra. 

In  naming  triangular  pyramids,  name  the  vertex  first, 
then  the  three  corners  of  the  base,  thus,  A. BCD.  Note  that 
any  corner  of  such  a  pyramid  may  be  regarded  as  a  vertex, 
the  other  three  being  the  corners  of  the  base  (just  as  any 
corner  of  a  A  may  be  regarded  as  a  vertex,  and  the  other 
two  corners  as  the  ends  of  the  base). 

A  problem  like  the  following  is  better  understood  if  a 
prism  is  actually  cut  up,  perhaps  a  wooden  one  made  in  the 
carpenter's  shop;  or  one  may  be  cut  neatly  from  a  bar  of 
soap.  It  is  easier  to  cut  up  the  latter  with  a  thin  knife 
than  to  cut  up  the  former  with  a  saw. 

A  suitable  model  for  showing  that  a  prism  is  equal  to 
three  times  the  volume  of  a  pyramid  on  the  same  base  and  of 
the  same  height  is  a  little  troublesome  to  make.  Inasmuch  as 
it  is  particularly  useful  in  the  demonstration  of  that  important 
principle,  we  give  a  few  hints  for  constructing  it. 

Fig.  124,  i,  represents  the  complete  triangular  prism, 
with  bases  ABC,  DBF.  From  it,  cut  the  pyramid  E.ABC 

(E291)  20 


290 


CRAFTSMANSHIP  IN  MATHEMATICS 


by  holding  the  knife  (or  saw)  at  E,  and  cutting  through  to 
AC.    iii  shows  the  pyramid  cut  off. 

Now  we  have  to  cut  the  remaining  piece  (ii)  into  two 
other  pyramids.     Cut  from  it  the  pyramid  C.DEF.     To  do 


this,  hold  the  knife  again  at  E,  and  cut  down  to  DC.  v  shows 
the  pyramid  cut  off,  its  new  face  being  shaded,  iv  shows 
the  part  left.  It  is  a  curious-looking  wedge-shaped  pyramid. 
We  will  name  it  E.DAC. 

We  may  show  (since  pyramids  on  equal  bases  and  of  the 
same  vertical  height  have  the  same  volume)  that  the  three  pyra- 
mids (iii,  iv,  v)  are  equal  in  volume.  The  bases  of  iii  and  v, 
E.ABC  and  C.DEF,  are  equal,  since  they  are  the  bases  of  the 
prism;  and  the  heights  are  equal,  for  FC  —  EB,  and  these  are 
two  of  the  long  edges  of  the  prism.  Again,  if  we  name  iv  and 
v  E.ACD  and  E.FDC,  we  see  that  the  bases  are  equal,  for 
they  are  the  halves  of  DACF,  one  of  the  faces  of  the  prism; 
and  their  vertical  heights  are  equal,  since  the  two  have  a 
common  vertex  E.  Hence  the  volumes  of  all  three  pyramids 
are  equal. 

It  is  a  simple  matter  to  make  the  "  nets  "  (fig.  125)  of 
the  three  pyramids,  and  fold  them  up  to  make  models.  The 
models  may  then  be  placed  together  to  form  the  prism.  Each 
net  will,  of  course,  consist  of  four  triangles,  the  sides  of  all 
of  which  will  be  edges,  or  diagonals  of  the  faces,  of  the  prism. 

The  formal  mensuration  of  geometry  of  the  pyramid,  then 
of  the  cone,  then  of  the  cylinder,  is  interesting  and  valuable, 


SOLID  GEOMETRY 


291 


Fig.  125 

and  I  do  not  find  that  it  gives  teachers  much  trouble,  always 
provided  that  the  necessary  preliminary  work  from  Euclid 
XI  on  lines  and  planes  has  been  done  well. 

The  calculation  of  the  areas  of  the  surfaces  of  solids  is 
also  simple,  including  even  the  surface  of  the  sphere,  provided 
that  suitable  figures  are  drawn. 


Euclid  XI 

All  the  essential  propositions  from  Euclid,  Book  XI,  are 
now  included  in  the  leading  schoolbooks  on  geometry. 
Most  boys  find  the  reasoning  easy  enough,  but  many  have 
great  difficulty  in  understanding  the  figures,  unless  models 
are  available  to  help  visualization. 

It  will  suffice  to  touch  upon  Euclid  XI,  4  and  6. 

XI,  4.  If  a  straight  line  is  perpendicular  to  each  of  two 
intersecting  straight  lines  at  their  point  of  intersection,  it  is 
perpendicular  to  the  plane 
containing  them. 

The  first  of  the  follow- 
ing figures  is  Euclid's 
own,  and  to  most  boys 
it  is  incomprehensible. 
The  second  is  that  found  Fig  I26 

in    many    modern    text- 
books.    This  is  a  case  where  a  model  is  certainly  desirable. 
Failing  that,  two  figures  should  be  drawn,  from  which  the 


292 


CRAFTSMANSHIP  IN  MATHEMATICS 


different  planes  may  easily  be  picked  out.  The  following 
figures  are  suitable:  in  the  first,  the  horizontal  plane  and  the 
vertical  planes  are  easily  seen;  in  the  second,  the  two  oblique 
planes.  If  such  figures  are  steadily  gazed  at,  with  one  eye, 


Fig.  127 

through  a  very  small  hole  in,  say,  a  piece  of  cardboard, 
they  quickly  assume  an  appearance  of  three  dimensions. 

XI,  6.  If  two  straight  lines  are  perpendicular  to  the  same 
plane,  they  are  parallel. 

The  first  figure  is  Euclid's  (again  a  poor  thing);  the 
second  is  that  commonly  found  in  school  textbooks.  In 


Fig.  128 

this  case,  again,  the  planes  want  sorting  out,  to  help  visual- 
ization. Figure  129  is  more  suitable,  with  the  horizontal 
plane  shaded.  The  two  perpendiculars  AB,  CD  are  shown 
by  rather  thicker  lines.  The  two  congruent  triangles  FDA, 
EDA  in  the  oblique  plane  AFE  are  easily  picked  out;  so 
are  the  two  BDE,  BDF  in  the  horizontal  plane.  But  it  is  so 
difficult  to  draw  a  figure  that  will  show,  to  a  beginner's  eye, 


SOLID  GEOMETRY  293 

the  two  congruent  triangles  ABE,  ABF  in  their  separate 
vertical  planes,  that  either  a  wire  model  or  a  pair  of  stereo- 
grams  are  certainly  desirable. 


Fig.  129 

It  is  unfortunate  that  so  many  boys  experience  difficulty 
in  visualizing  three-dimensional  figures.  But  the  fact  has 
to  be  allowed  for,  and  provision  made  accordingly. 

Do  not  press  too  far  the  argument  that  such  aids  as  models 
should  be  withdrawn  in  order  that  the  boys'  imagination  may 
be  given  opportunity  to  develop.  The  boys'  developed 
imagination  will  be  a  poor  thing  if  it  has  to  be  nurtured  on 
the  teacher's  badly-drawn  figures. 


CHAPTER  XXII 

Orthographic  Projection 

Elementary  Work 

Below  we  reproduce  subject-matter  suitable  for  two  or 
three  preliminary  lessons  on  orthographic  projection  to  the 
Middle  Forms.  Time  can  seldom  be  found  for  much  ruler 
and  compass  work,  but  freehand  drawings,  rapidly  executed 


294 


CRAFTSMANSHIP  IN  MATHEMATICS 


in  association  with  the  teacher's  own  blackboard  demon- 
strations, may  be  made  to  serve  a  useful  purpose  in  laying 
the  foundations  of  the  subject.  Higher  up  the  school,  if  time 
permits,  more  advanced  work  should  be  taken.  It  helps  the 
ordinary  geometry,  plane  and  solid,  greatly. 

In  preparing  drawings  for  builders,  architects  make  plans 
and  elevations  of  buildings  to  be  erected.  A  plan  of  a 
thing  is  an  outline  on  a  horizontal  plane;  an  elevation  is  an 
outline  on  a  vertical  plane. 

Push  the  table  up  against  the  wall.  On  the  table  place 
a  rectangular  block  with  two  faces  parallel  to  the  wall.  Chalk 

on  the  table  an  outline  of 
the  base  of  the  block,  and 
thus  make  a  plan  of  the 
block.  Now  push  the  block 
against  the  wall,  and  chalk 
an  outline  on  the  wall. 
This  is  an  elevation  of  the 
block;  the  elevation  a'b'c'd' 
in  the  figure  is  a  projec- 
tion of  the  face  abed  of  the 
block. 

It  is  a  very  simple  kind 
of  projection,  because  most  of  the  work  to  be  done  depends 
on  the  drawing  of  perpendiculars  and  parallels.  The 
projectors  and  other  working  lines  are  nearly  all  perpen- 
diculars and  parallels.  A  word  implying  perpendiculars 
and  parallels  is  "  orthographic  ",  and  the  projection  is  some- 
times called  orthographic  projection. 

As  it  is  not  very  convenient  to  draw  on  the  wall,  we 
sometimes  use  two  boards  hinged  at  right  angles.  The  next 
figure  shows  such  a  pair,  first  of  all  hinged  in  position,  then 
unhinged  and  the  vertical  plane  turned  back  into  the  hori- 
zontal. The  figure  shows  two  plans  and  elevations  of  the  hut 
in  fig.  136.  The  first  plan  shows  the  long  sides  of  the  hut 
parallel  to  the  vertical  plane  and  the  elevation  a  side  eleva- 


Fig.  130 


ORTHOGRAPHIC  PROJECTION 


295 


tion.  The  second  plan  shows  the  long  sides  perpendicular 
to  the  vertical  plane,  and  the  elevation  an  end  elevation. 
The  term  front  elevation  is  also  sometimes  used.  An  ele- 
vation is  often  spoken  of  as  a  view. 

An  architect  would  not  draw  two  plans  of  one  building, 
but  he  would  always  draw  two  or  more  elevations,  in  order 


Fig.  131 

to  make  the  builder  understand  exactly  what  the  building 
was  to  be  like. 

Note  that  all  the  projectors  (shown  as  broken  lines)  are 
perpendiculars  and  parallels. 

The  two  boards  are  shown  hinged  merely  to  help  you  to 
understand  how  the  plan  and  elevation  are  related,  but  plans 
and  elevations  are  commonly  drawn  as  in  fig.  ii.  An  architect 
would  not  trouble  to  draw  the  outlines  of  the  two  boards. 
He  just  rules  a  line  across  the  paper  (marking  it  XY  some- 
times), draws  the  plan  below,  and  the  elevation  above.  More 
frequently  than  not,  he  uses  separate  pieces  of  paper,  but 
he  always  remembers  how  the  separate  drawings  are  related. 
The  XY  line  is  sometimes  called  the  ground  line:  it  is  the 
line  of  contact  of  the  vertical  plane  with  the  ground.  It  is 
usual  to  keep  the  plan  a  little  distance  away  from  this  line, 
but  to  let  all  elevations  stand  on  the  line. 

Although  your  plans  and  elevations  will  always  be  drawn 
on  the  flat,  you  will  sometimes  find  it  useful  to  fold  your 
paper  at  right  angles  on  the  XY  line,  and  to  place  the  object, 
if  small  enough,  in  position  on  the  horizontal  plane.  You 
can  then  see  more  plainly  what  the  elevation  on  the  vertical 


296 


CRAFTSMANSHIP  IN  MATHEMATICS 


plane  will  be.  For  instance,  in  the  figure  at  the  beginning 
of  this  section,  suppose  a  pencil  is  placed  in  contact  with 
the  edge  ad  of  the  block,  and  the  block  removed.  You  can 
see  at  once  that  the  plan  of  the  pencil  is  bcy  and  that  the 
elevation  is  a'd'. 

Remember  that  plans  and  elevations  of  any  object  are 
obtained  by  drawing  perpendicular  projectors  to  the 
Horizontal  Plane  (H.P.)  and  Vertical  Plane  (V.P.).  The 
feet  of  these  projectors  are  then  joined  in  such  a  way  that 
the  lines  correspond  to  the  edges  of  the  object  itself. 

Here  are  some  examples  of  plans  and  elevations.  Copy 
them  full  size.  Then  fold  your  paper  on  the  ground  line 
and  turn  the  V.P.  into  position. 

1.  Plans  and  elevations  of  a  line  3"  long.  Hold  a  piece 
of  wire  or  a  short  pencil  in  position,  so  that,  looked  at  from 


x — j- 


567 


Fig.  132 


above,  it  covers  the  plan,  and  looked  at  from  the  front,  it 
covers  the  elevation.  Then,  in  each  case,  try  to  describe 
the  position  of  the  pencil  with  reference  to  the  two  planes, 
checking  your  descriptions  by  the  correct  descriptions  below. 


i. 

2 

3. 

4. 

5. 

6. 

7. 

To  the  H.P.  the  line  is 

II 

JL 

II 

60° 

|| 

45° 

60° 

To  the  V.P.  the  line  is 

II 

II 

J_ 

II 

45° 

45° 

30° 

2.  Plans  and  elevations  of  a  rectangular  sheet  of  paper, 
3"  X  2": 


ORTHOGRAPHIC  PROJECTION 


297 


The  Plane  of 
the  Paper  is 

The  Long 
Edges  are 

The  Plane  of  the 
Paper  is 

The  Long  Edges  are 

1. 

||  to  V.P. 

||  to  H.P. 

5. 

||  to  V.P. 

30°  to  H.P. 

2, 

II      V  P 

JL  „  H.P. 

0 

II   „  H.P. 

60°  „  V.P. 

3. 

II  ,,  H.P. 

II  „  V.P. 

7. 

45°  „  both  planes 

||    „  both  planes 

4. 

II  „  H.P. 

JL  „  V.P. 

8. 

-L    ,,  both  planes 

45°  „  both  planes 

3.  Plans  and  elevations  of  a  square  prism: 


Positions  as  follows: 

(1)  Standing  on  base,  two  sides  ||  to  V.P. 

(2)  Lying  on  a  side,  all  sides  JL  to  V.P. 

(3)  Lying  on  a  side,  bases  J-  to  both  planes. 

(4)  Standing  on  a  base,  one  diag.  of  base  J-  to  V.P. 

(5)  Lying  on  a  side,  two  sides  30°  with  V.P. 

(6)  Same  as  No.  1,  with  a  section  AB  ±  to  V.P. 

(7)  Same  as  No.  5,  with  a  section  CD  JL  to  H.P. 


298 


CRAFTSMANSHIP  IN  MATHEMATICS 


It  is  sometimes  necessary  to  know  the  shapes  of  sections 
of  solids.  Two  are  shown  above.  The  cut  surfaces  are  in- 
dicated by  cross  hatching.  When  in  doubt  about  such  a 
shape,  make  a  rough  model  and  cut  through  it. 

The  positions  of  objects  with  respect  to  the  two  planes 
may  be  described  in  more  than  one  way.  In  the  first  of  the 
last  series,  for  instance,  we  might  have  said  two  sides  JL  to 
V.P.  In  the  third,  we  might  have  said  two  sides  ||  to  the 
V.P.  and  two  ||  to  H.P.  But  the  description  must  always  be 
sufficient  to  fix  the  object  in  a  particular  position. 

4.  Plans  and  elevations  of  other  solids: 


Fig.  135 


Position  of  solids: 


(1)  Hexagonal  prism  standing  on  base,  two  sides  ||  to  V.P. 

(2)  Hexagonal  prism  lying  on  a  side,  long  edges  30°  to  V.P. 


ORTHOGRAPHIC  PROJECTION  299 

(3)  Hexagonal  pyramid  standing  on  base,  two  edges  of 
base  ||  to  V.P. 

(4)  Cylinder  standing  on  base,  with  section  AB  -L  to  H.P. 

(5)  Cylinder  lying  down,  axis  30°  to  V.P. 

(6)  Cone  standing  on  base,  with  section  CD  -L  to  V.P. 

Sections  which  are  cut  obliquely  to  one  of  the  two  pro- 
jection planes  may  give  a  little  trouble,  especially  if  their 
shapes  cannot  be  first  imagined.  The  shape  of  an  oblique 
section  through  a  cylinder  may  be  shown  by  half  filling  a 
round  bottle  with  water  and  holding  the  bottle  obliquely; 
the  water-surface  gives  the  shape  of  the  section — an  ellipse. 
So  with  a  square  bottle,  or  a  conical  flask.  Or  you  may  push 
the  solid  obliquely  into  the  ground,  down  to  the  level  of  the 
section  line.  The  shape  of  the  mouth  of  the  hole  is  the  shape 
of  the  section. 

More  Advanced  Work 

Here  are  types  of  problems  suitable  for  more  advanced 
pupils: 

1.  Determine  the  projection  of  three  spheres  of  different 
radii,  resting  on  the  ground  in  mutual  contact. 

2.  Determine  the  projections  of  the  curve  of  intersection 
of  a  cone  penetrating  a  cylinder,  the  axes  of  the  two  solids 
intersecting  at  a  given  angle. 

3.  Determine  the  shadow  cast  by  the  hexagonal  head  of 
a  bolt  with  a  cylindrical  shaft,  the  bolt  standing  vertically 
on  its  screw  end,  from  given  parallel  rays. 

4.  Determine  the  shadow  cast  by  a  cone  standing  on  the 
ground,  the  direction  of  the  light  being  so  arranged  as  to 
throw  part  of  the  shadow  on  the  vertical  plane. 

For  shadow-casting  problems,  it  is  a  good  plan  to  place 
the  object  in  strong  sunlight,  so  that  the  shadow  can  actually 
be  cast  on  the  horizontal  plane  (and  vertical  plane,  too,  if 
necessary),  and  examined.  The  problems  then  become  very 


300  CRAFTSMANSHIP  IN  MATHEMATICS 

simple.    Shadows  cast  by  artificial  lights  are  less  serviceable, 
since  the  light  rays  are  necessarily  not  parallel. 

As  a  rule  there  is  no  time  for,  and  there  is  very  little  point 
in,  making  projections  of  groups  of  objects,  but  cases  of  simple 
interpenetrations  make  such  good  problems  that  one  or  two 
are  worth  doing. 

Speaking  generally,  the  ground  covered  in  orthographic 
projection  should  be  enough  to  enable  Sixth  Form  boys 
to  solve,  readily  and  intelligently,  such  stock  theorems  of 
projection  as  these: 

The  projection  on  a  plane  of  an  area  in  another  plane; 
and  particular  cases,  e.g.: 

(a)  Projection  of  an  ellipse  into  a  circle,  and  the  ratio 
of  their  areas. 

(f$)  The  projective  correspondence  between  the  per- 
pendicular diameters  of  a  circle  and  conjugate  diameters  in 
an  ellipse. 

(y)  Extension  of  the  properties  of  polars  from  the  circle 
to  the  ellipse. 


CHAPTER   XXIII 

Radial   Projection 

First  Notions 

For  Sixth  Form  boys  learning  mathematics  seriously,  a 
knowledge  of  radial  projection  is  at  least  as  important  as  a 
knowledge  of  orthographic  projection.  Here  is  the  sub- 
stance of  a  lesson  for  beginners:  it  is  the  sort  of  lesson  one 
might  expect  to  hear  an  intelligent  art  teacher  give. 

Stand  about  18"  or  24"  from  a  window,  keeping  your 
head  perfectly  steady,  and,  with  a  piece  of  wet  chalk,  trace 
accurately  on  the  glass  an  outline  of  a  distant  building. 


RADIAL  PROJECTION  301 

When  you  have  finished,  it  is  easy  to  imagine  straight  threads 
passing  from  all  the  principal  points  in  the  building,  through 
the  corresponding  points  in  your  sketch  on  the  glass,  to 
your  eye.  Every  line  in  the  sketch  exactly  covers  the  corre- 
sponding line  in  the  building.  The  drawing  is  another  kind 
of  projection.  But  the  projectors  are  no  longer  perpen- 
diculars; they  all  radiate  from  your  eye,  and  they  all  pass 
through  the  vertical  plane  on  which  you  have  made  the 
sketch,  to  the  building.  The  vertical  plane  on  which  you 
have  made  the  picture  is  called  the  picture  plane.  This 
kind  of  projection  is  called  radial  projection  or  perspective 
projection.  Perspective  drawings  are  the  kind  of  drawings 
made  by  artists.  Pictures  are  painted  in  accordance  with 
the  rules  of  perspective.  The  camera  also  follows  these 
rules.  Pictures  and  photographs  represent  things  as  they 
appear  to  the  eye. 

Here  is  the  perspective  projection  of  a  hut: 


In  the  hut  itself,  the  three  vertical  lines,  AB,  CD,  EF,  are 
all  equal.  In  the  drawing,  CD,  the  one  nearest  the  observer, 
is  the  longest,  and  those  farther  away  are  shorter.  So  with 
the  verticals  in  the  doors  and  in  the  windows.  All  parallels 
which  recede  from  the  observer  seem  to  approach  each  other, 
and  at  last  to  meet  at  a  point  on  a  line  level  with  the  eye. 
Equal  parts  of  a  horizontal  in  the  object  are  unequal  in  the 
drawing  (compare  the  horizontal  window-bars  in  the  two 
windows).  The  farther  away  a  thing  is  taken,  the  shorter  it 
becomes  in  the  drawing.  If  you  have  made  an  accurate 
chalk-drawing  on  the  window,  you  can  teach  yourself  a  good 
deal  about  perspective. 


302 


CRAFTSMANSHIP  IN  MATHEMATICS 


If,  however,  your  chalk-drawing  is  not  satisfactory,  do 
this  instead.  Take  a  rectangular  wooden  frame  of  some  sort 
(an  old  picture-frame  will  do),  say  about  15"  X  10".  Drive 
in  tacks  two-thirds  of  their  length,  at  equal  distances  apart, 
say  1",  all  round  the  edge.  Stretch  cotton  across  the  frame 
and  round  the  tacks  in  such  a  way  as  to  divide  up  the  frame 
into  squares.  Now  divide  up  a  piece  of  drawing  paper  into 
the  same  number  of  squares.  Place  the  frame  in  a  vertical 
position  between  your  eye  and  a  suitable  object  or  view 
that  may  be  sketched.  If  you  sketch  a  house  (a  very  suitable 
object)  get  back  far  enough  to  see  the  whole  house  easily 
within  the  frame.  Now  observe  what  part  of  the  object 
appears  within  a  particular  square  of  the  frame,  and  sketch 
that  part  in  the  corresponding  square  on  your  paper.  And 
so  on.  With  care  you  may  make  a  fairly  accurate  drawing, 
and  can  then  learn  a  good  deal  about  perspective,  more 
particularly  about  converging  lines  and  diminishing  lengths. 
You  may  also  learn  much  from  a  large  photograph  of  a  build- 
ing, especially  if  you  can  compare  the  photograph  with  the 
building  itself. 


Here  is  a  perspective  sketch  of  three  bricks  in  a  row.    It 
is  as  they  would  appear  in  a  photograph.    The  middle  brick 


Fig.  138 


is  in  the  middle  of  the  picture,  and  the  photographer  points 
his  camera  towards  it.  If  he  were  photographing,  say,  brick 
C  alone,  he  would  turn  his  camera  round  and  point  towards 


RADIAL  PROJECTION 

that  brick.  His  picture  would 
then  be  like  D.  Neither  A  nor  C 
is  the  correct  drawing  of  a  brick 
unless  the  brick  is  to  the  left  or 
right  of  a  group  of  things,  as  in 
fig.  137. 


303 


The  Picture  Plane. 
Models 


Use  of 


The  ordinary  perspective  text- 
book prepared  for  Art  teachers  is, 
generally,  just  a  book  of  rules, 
rules  with  only  the  faintest  tinge 
of  mathematics  in  them.  I  have 
known  boys  make  faultless  and 
most  elaborate  perspective  draw- 
ings of  groups  of  objects  in  dif- 
ferent positions,  and  yet  they  have 
had  the  most  hazy  ideas  of  the 
inner  nature  of  the  rules  they  have 
been  applying.  And  yet,  at  bottom, 
the  whole  thing  is  a  study,  and  a 
simple  study,  too,  of  similar  tri- 
angles. 

This  figure  shows  the  Picture- 
plane  12  ft.  from  the  observer, 
with  his  eye  12  ft.  away  and  5  ft. 
above  the  ground  at  S.  P.  (his 
Station  Point).  The  Picture-plane 
meets  the  ground  in  the  ground- 
line.  The  point  on  the  Picture- 
plane  immediately  opposite  the 
eye  is  the  Centre  of  Vision.  The 
horizontal  through  the  Centre  of 
Vision  is  called  the  Horizontal 
Line.  Radial  projectors  run  from 


304 


CRAFTSMANSHIP  IN  MATHEMATICS 


the  eye  to  each  corner  of  a  block  fixed  behind  the  Picture- 
plane   and   cut   the   Picture-plane    in  points   which,   when 


RADIAL  PROJECTION  305 

joined  up,  give  on  the  Picture-plane  a  perspective  picture  of 
the  block. 

Fig.  140  shows  the  sort  of  perspective  drawing  that  appears 
in  the  textbooks.  The  pupil  must  see  the  relation  between 
figs.  139  and  140.  In  fig.  139  the  line  from  the  eye  to  the 
C.  of  V.  is  represented  at  right  angles  to  the  P.P.  That  line 
must  be  supposed  to  be  hinged  at  C.  of  V.  and  to  turn 
on  the  hinge  through  90°  until  it  comes  into  the  same  plane 
as  the  P.P.,  as  in  fig.  140,  which  represents  a  drawing  in  one 
plane,  the  plane  of  the  paper.  It  is  imperative  that  the  boys 
see  fig.  139  as  a  model.  Only  then  will  they  be  able  to  under- 
stand fig.  140  completely.  Then  the  points  of  distance, 
vanishing  points,  and  measuring  points  are  all  a  matter  of 
very  easy  geometry. 

In  practice,  it  is  an  advantage  to  substitute  for  the  glass 
P.P.  a  sheet  of  perforated  zinc,  or  a  square  of  stretched 
black  filet  net  of  TV'  mesh,  and  to  run  threads  (fastened 
with  drawing  pins  to  the  corner  of  the  rectangular  block  or 
other  object  being  sketched)  through  the  appropriate  holes 
in  the  zinc  or  net  to  the  ring  representing  the  eye,  where 
they  may  be  secured.  A  drawing  may  then  be  represented 
in  threads  of  another  colour,  run  from  hole  to  hole  in  the 
zinc  or  net,  instead  of  in  chalk  as  when  glass  is  used. 

Main  Principles 

The  main  principles  of  perspective,  mathematically 
considered,  are  all  reducible  to  a  small  handful  of  three- 
dimensional  problems.  One  will  suffice  to  illustrate  the 
degree  of  difficulty. 

Given  any  point  on  the  ground-plane ',  to  determine  its 
position  on  the  picture-plane. 

Since  solids  are  determined  by  planes,  planes  by  lines, 
and  lines  by  points,  it  will  suffice  to  determine  the  position 
in  the  picture-plane  of  just  one  point  on  the  ground-plane. 
This  really  solves  the  general  problem,  inasmuch  as  any 
other  point  may  be  similarly  determined. 

(E291)  31 


306  CRAFTSMANSHIP  IN  MATHEMATICS 

Let  M  be  the  point  on  the  ground-plane.  Drop  a  per- 
pendicular MN  on  the  picture-plane,  and  another  EC  from 
the  eye  E  to  the  C.  of  V.  on  the  P.P.  Since  EC  is  parallel 
to  MN,  both  EC  and  MN,  and  also  CN  and  ME,  are  in  the 
shaded  oblique  plane.  NC  is  the  complete  projection  of 
the  line  NM  extended  to  an  unlimited  length  behind  the 


Fig.  141 

P.P.;  and  the  point  R,  where  ME  intersects  CN,  is  the  pro- 
jection of  M.    Hence  R  is  the  point  to  be  determined. 

Now  in  the  oblique  plane  we  have  the  two  similar  triangles 

NR        MN 

RCE,  RMN,     Hence  —  =  tEl9  i.e.  CN  is  divided  at  R 
.      ,          .  RC        CE 

in  the  ratio: 

distance  of  point  M  from  P.P. 
distance  of  observer  from  P.P. 

Thus  CN  being  drawn, 
R  can  be  determined  at 
once. 

Suppose  M  is  3' 
behind  the  P.P.  and  the 
observer  is  10'  in  front. 
It  is  required  to  divide 
CN  in  the  ratio  3  :  10. 

Through  C  and  N  draw  any  pair  of  parallels.    Measure  off 
NX  equal  to  3  units  and  CY  equal  to  10  units.    We  have 


N 


Fig.  142 


RADIAL  PROJECTION  307 

.    .,          .      ,  „  NR      NX       3      .         . 

two    similar   triangles.       Hence    ~—  =  — —  =  — ,    i.e.    the 

position  of  R  in  CN  is  determined. 

Hence  if  the  space  between  the  two  parallels  represents 
the  P.P.,  if  C  is  the  C.  of  V.,  and  if  CN  is  the  complete  projec- 
tion of  a  line  perpendicular  to  the  picture-plane  and  meeting 
it  in  N,  the  projection  of  any  point  in  this  perpendicular  line 
may  be  found  by  the  above  method. 

In  practical  perspective  we  use  as  a  pair  of  parallels  the 
horizontal  eye-line  and  the  ground  line.  This  is  a  mere 
matter  of  convenience;  any  other  pair  of  parallels  drawn  on 
the  P.P.  would  do  equally  well. 

Measuring  points,  vanishing  points,  and  the  rest  are  all 
determined  by  the  consideration  of  virtually  the  same  prin- 
ciple. In  fact  the  complete  art  of  perspective  projection 
lies  in  that  principle.  With  the  model,  the  whole  thing  be- 
comes simplicity  itself.  The  perforated  zinc  (or  net)  P.P. 
with  strings  passing  through  to  the  eye,  and  the  projection 
of  the  figure  threaded  in  with  threads  of  a  different  colour, 
make  the  main  principles  so  clear  that  there  is  little  need 
for  formal  demonstration.  The  similar  triangles  then  in 
situ  tell  the  whole  story. 

Sixth  Form  Work 

When  these  main  principles  underlying  the  practice  of 
perspective  have  been  mastered,  the  subject  should  be 
followed  up  in  the  Sixth  by  a  few  of  the  stiffer  propositions 
associated  with  the  general  theory  of  perspective,  treated 
formally  and  deductively,  more  especially  those  concerned 
with  triangles  in  perspective,  so  far  as  these  are  necessary 
for  the  understanding  of  the  chief  properties  of  the  hexastigm 
in  a  circle;  at  least  Pascal's  theorem  should  be  known,  though 
as  a  mere  fact  in  practical  geometry  this  theorem  should  be 
known  lower  down  the  school;  its  later  theoretical  considera- 
tion is  always  a  delight  to  the  k^en  mathematical  boy. 

But  to  attack  such  theorems  of  perspective  before  some 


308  CRAFTSMANSHIP  IN  MATHEMATICS 

understanding  of  the  practice  of  perspective  has  been  acquired 
is  to  attack  theorems  that  are  lifeless. 


CHAPTER  XXIV 

More  Advanced  Geometry 

A  Possible  Outline  Course 

What  is  sometimes  called  "  Modern  "  Geometry  or 
"  Pure  "  Geometry  usually  occupies  a  subordinate  position 
in  Sixth  Form  work.  This  is  to  be  regretted. 

It  may  be  readily  admitted  that  analysis  is  a  powerful 
instrument  of  research,  and  doubtless  for  this  reason  alone 
mathematicians  have  given  it  a  very  important  place  in 
recent  years.  Accordingly,  Sixth  Form  boys  tend  to  devote 
much  time  to  preparation  for  the  work  of  that  kind  which  is 
demanded  of  them  at  the  University.  But  it  cannot  be  denied 
that  an  intimate  acquaintance  with  geometry  is  only  to  be 
obtained  by  means  of  "  pure  "  geometrical  reasoning.  In 
the  classroom  no  branch  of  mathematics  is  so  productive 
of  sound  reasoning  as  is  pure  geometry.  The  ordinary 
geometrical  theorem  admits  of  a  simple,  rigorous,  and  com- 
pletely satisfactory  proof,  a  proof  that  is  convincing  and  not 
open  to  question.  An  elementary  knowledge  of  the  properties 
of  lines  and  circles,  of  inversion,  of  conic  sections,  treated 
geometrically,  of  reciprocation,  and  of  harmonic  section, 
ought  to  be  expected  from  all  Second  Year  Sixth  Form  boys. 
Many  boys  now  leave  school  without  any  conception  of  some 
of  the  remarkable  properties  of  the  triangle  and  circle;  and 
this  ought  not  to  be. 

There  is  much  to  be  said  for  beginning  with  rectilineal 
figures,  including  a  fairly  complete  study  of  the  tetragram 
and  tetrastigm,  the  more  elementary  properties  of  the  polygram 


MODERN  ADVANCED  GEOMETRY  309 

and  polystigm,  and  then  for  proceeding  with  harmonic  section. 
The  remaining  topics  follow  simply.  We  outline  for  teaching 
purposes  one  or  two  of  the  different  subjects. 

The  Polygram  and  Polystigm 

These  may  be  regarded  either  as  systems  of  lines  inter- 
secting in  points,  or  as  systems  of  points  connected  by  straight 
lines.  The  simplest  figure  is  that  determined  by  3  lines  or 
3  points.  If  we  have  any  3  lines  which  are  not  concurrent, 
or  if  we  have  any  3  points  which  are  not  collinear  and  which 
may  therefore  be  connected  by  3  straight  lines,  we  have 
two  systems  which  are  virtually  the  same,  and  we  may  give 
the  name  triangle  to  either. 

But  with  more  than  3  lines  or  points,  the  resulting  figures 
though  closely  related  are  not  identical. 

Rectilineal  figures  regarded  as  systems  of  lines  are  called 
polygrams;  as  systems  of  points,  polystigms. 

A  tetragram  in  its  most  general  form  is  a  complete  recti- 
lineal figure  of  four  lines  > 
no  3  of  which  are  con- 
current, and  no  2  parallel. 
Each  line  is  therefore  in- 
tersected by  the  other  3. 
If  the  lines  be  named  a, 
bt  c,  d,  their  points  of  in- 
tersection may  be  named 
by  combining  the  2  letters 
which  denote  the  inter- 
secting lines.  Since  there  '  Fig.  i43 
are  3  points  of  intersec- 
tion in  each  of  the  4  lines,  we  seem  to  have  12  points  of 
intersection  in  all,  but  these  are  reduced  to  6,  since  each 
is  named  twice.  The  6  points  of  intersection  are  called 
vertices. 

A  tetrastigm  in  its  most  general  form  consists  of  four 
primary  points,  no  3  of  which  are  collinear  and  which  do 


310  CRAFTSMANSHIP  IN  MATHEMATICS 

not  fall  in  pairs  in  parallel  lines.  If  the  points  be  named 
A,  B,  C,  D,  their  connectors  may  be  named  in  the  usual  way, 
AB,  BC,  &c.  Since  there  are  3  connecting  lines  terminating 

in  each  of  4  primary  points,  we  seem 
to  have  12  connecting  lines  in  all, 
but  these  reduce  to  6,  since  each  is 
named  twice.  The  6  lines  are  called 
connectors. 

From  suitable  figures,  the  number 
of   vertices    and    connectors    in    the 
pentagram  and  pentastigm  is  seen  to 
Fig.  i44  be  10,  and  in  the  hexagram  and  hexa- 

stigm,  15. 

We  infer  that  in  a  polygram  of  n  lines,  and  in  a  polystigm 
of  n  points,  the  number  of  vertices  and  connectors  are,  re- 
spectively, -~F-^;  for  the  tetragram  and  tetrastigm  give  us 
^-~>,  the  pentagram  and  pentastigm  '^p,  and  the  hexagram 
and  hexastigm  *-~. 

In  a  tetragram  a  diagonal  may  be  drawn  from  each  of 
the  vertices  to  another  vertex;  the  6  diagonals  reduce  to  3. 


Tetragram,  with  its  3  diagonals  Tetrastigm,  with  its  (4  primary  and) 

Fig.  145  3  secondary  points 

In  a  tetrastigmy  each  of  the  6  connectors  can  intersect 
another  connector  at  a  point  other  than  the  4  primary  points; 
the  6  reduce  to  3.  These  3  new  points  are  called  the  secondary 
points  of  the  tetrastigm. 

From   suitable   figures   the   number   of  diagonals   and 


MODERN  ADVANCED  GEOMETRY      311 

secondary  points  in  the  pentagram  and  pentastigm  is  seen 
to  be  15,  and  in  the  hexagram  and  hexastigm,  45. 
Since 

0        4x3x2x1,-        5x4x3x2     AK       6x5x4x3 

&  =  15  =  45  = 

8  8  8 

we  infer  that  in  a  polygram  of  n  lines,  and  in  a  polystigm 
of  n  points,  the  number  of  diagonals  and  of  secondary  points 
respectively  is  *("-i)(*-2)  <"-*>. 

The  pupil  should  check  for  the  pentagram  and  pentastigm. 
The  figures  for  the  hexagram  and  hexastigm  are  complicated 
and  their  analysis  is  hardly  worth  while.  The  pupil  should 
note  that  a  polygram  and  polystigm  of  the  same  order  are 
reciprocal  figures;  they  give  us  an  excellent  example  of  the 
principle  of  duality. 

Derived  Polygons 

The  pupil  may  be  encouraged  to  establish,  from  an  ex- 
amination of  a  few  particular  cases,  the  principle  that  the 
number  of  derived  polygons  from  a  polygram  or  polystigm 
is  fe^L'. 

For  instance,  the  number  of  derived  tetragons  from  a 
tetragram  or  tetrastigm  is  3x  ^x  1  =  3;  of  derived  pentagons 
from  a  pentagram  or  pentastigm  is  4x3*2x  1  —  12;  of  hexa- 
gons, 60;  and  so  on. 

Here  is  a  tetragram  and  its  3  derived  tetragons: 

at> 


A  tetragram  consists  of  4  lines  with  6  consequent  vertices, 
and  3  vertices  lie  on  each  of  the  4  lines.    But  in  a  tetragon 


312  CRAFTSMANSHIP  IN  MATHEMATICS 

there  are  only  2  vertices  in  a  line,  viz.  those  at  the  extremities 
of  the  line;  there  are  thus  4  vertices  in  all.  Hence,  for  a 
derived  tetragon,  we  have  to  select  4  vertices  out  of  the  6, 
in  such  a  way  that  2,  and  2  only,  may  lie  on  and  determine 
the  extremities  of  each  of  the  4  lines.  Such  a  selection  is 
known  as  a  complete  set  of  vertices  for  a  derived  tetragon. 
Note  that,  whatever  vertex  is  chosen  as  a  starting-point,  that 
vertex  must  be  the  point  where  the  figure  is  completed. 
Here  is  a  tetrastigm  and  its  3  derived  tetragons. 


Fig.  147 

Analogous  reasoning  applies.  We  have  to  select  4  connectors 
out  of  6,  in  such  a  way  that  2,  and  only  2,  may  terminate  in 
each  of  the  4  vertices.  The  selection  is  known  as  a  complete 
set  of  connectors  for  a  derived  tetragon. 

The  boys  may  be  given  the  task  of  drawing  the  12 
pentagons  from  a  pentagram  and  the  12  from  the  pentastigm. 
But  they  must  set  to  work  systematically  or  there  will  be 
confusion.  Consider  the  pentastigm  with  its  5  primary 
points,  A,  B,  C,  D,  E.  Select  AB  as  the  initial  connector. 
Associated  with  it  as  a  second  connector  we  may  have  BC,  BD, 
or  BE;  we  then  have  the  first  two  connectors  formed  in  3 
different  ways,  viz.  AB,  BC;  AB,  BD;  AB,  BE.  The  first 
two  connectors  being  fixed,  the  remaining  3  can  be  selected 
in  2  different  ways,  and  thus  we  have  6  different  pentagons 
formed  with  AB  as  a  first  connector.  Now  do  exactly  the 
same  thing  with  the  other  3  connectors  terminating  in  A. 
And  so  on. 

Do  not  forget  that  the  polystigm  is  the  key  to  many  of 
the  mediaeval  tree-planting  problems.  Given  n  trees,  what 
is  the  greatest  number  of  straight  rows  in  which  it  is  possible 


MODERN  ADVANCED  GEOMETRY 


313 


to  plant  them,  each  row  to  consist  of  m  trees?     For  instance, 
given  16  trees,  plant  them  in  15  rows 
of  4. 

Construct  a  regular  pentagram  with 
such  of  its  diagonals  as  are  necessary  to 
form  an  inner  second  pentagram.  The 
introduction  of  these  diagonals  gives  6 
new  points,  which,  with  the  10  vertices 
of  the  pentagram,  make  16  points. 

The  general  problem  has  never 
been  completely  solved. 


Harmonic  Division  * 

The  Pythagoras  relation,  golden  section,  and  harmonic 
division,  are  the  3  keys  of  pure  geometry,  yet  harmonic  division 
frequently  receives  but  very  scant  attention.  The  principle 
itself  once  fully  grasped,  the  actual  proofs  of  theorems  in- 
volving it  are  generally  of  the  simplest. 

The  approach  to  the  subject  and  its  problems  may  be 
effectively  made  in  this  way: 

(1)  Divide  a  line  internally  and  externally  in  the  same  ratio, 


Fig.  149 


say  5:2.    Note  that  the  correct  reading  of  the  ratios  is  from 
the  extremities  of  the  line  to  the  point  of  division;   thus  for 


•In  speaking  of  cross-ratios,  avoid  the  term  " anharmonic ",  since  it  implies 
"  not  harmonic  ",  whereas  a  cross-ratio  may  be  harmonic,  for  it  may  be  the  cross- 
ratio  of  an  harmonic  range. 


314  CRAFTSMANSHIP  IN  MATHEMATICS 

internal  division  we  have  — ~ ;  for  external  division  — — -.  Also 

BPj  BP2 

note  the  sign  as  well  as  the  magnitude;  e.g.  for  internal 
division,  APl  and  BPX  are  measured  in  opposite  directions 
and  the  ratio  is  therefore  negative;  for  external  division 
AP2  and  BP2  are  measured  in  the  same  direction,  and  the 
ratio  is  therefore  positive.  Give  important  examples  of  this, 
for  instance  the  theorems  of  Ceva  and  Menelaus. 

(2)  Algebraic   Harmonic   Progression. — Definition:     If  a, 

,         ,  .     TT  „      ,        a  —  b       a  1.12 

0,  and  c  are  in  H.P.,  then =  -;     or,    -  +  -  —  -;    or 

0  b  —  c       c  a       c       b 

h  — 

a  -f  c' 

(3)  Compare  the  geometry  and  the  algebra. — A  line  AQ 
is  said  to  be  "  harmonically  divided  "  at  P  and  B  when,  if 
AQ  =  a,  AB  =  b9  AP  =  c,  a,  b,  and  c  are  in  H.P. 

c.          a  -  b      a  /t      ,  c  .  .     ,         .    BQ      AQ 

Since =  -  (by  definition),       /._=:_, 

b  —  C         C  JjJr         Ar 

or    AQ  X  BP  =  AP  X  BQ;  (Cf.  fig.  150.) 

i.e.  product  of  whole  line  and  middle  segment  equals  product 
of  external  segments.  Hence  if  AB  is  divided  harmonically 


A  I 1 ! 19 

|^ .j-a-J *, 

j^_ b 1 -^j 

K--c H 

Fig.  150 

at  P  and  Q,  PQ  is  divided  harmonically  at  A  and  B.  In  other 
words,  AB  is  divided  internally  and  externally  at  P  and  Q 
in  the  same  ratio;  and  PQ  is  divided  internally  and  externally 
at  B  and  A  in  the  same  ratio. 

(4)  Harmonic  Ranges. — If  a  line  AB  is  divided  harmonically 
at  P  and  Q,  the  range  of  points  {AB,  PQ}  is  called  a  harmonic 
"  range  ".  The  pair  of  points  A  and  B  are  said  to  be  con- 
jugate to  each  other;  so  with  the  points  P  and  Q.  We  may 


MODERN  ADVANCED  GEOMETRY  315 

conveniently  name  a  harmonic  range  thus  (AB,  PQ},  the 
comma  being  inserted  to  distinguish  the  pairs  of  conjugate 
points. 

(5)  Harmonic  Pencils. — Define  "  ray  "  and  "  pencil  ". 
Every  section  of  a  harmonic  pencil  is  a  harmonic  range,  e.g. 
(AB,  PQ),  (A'B',  P'Q').  A  pencil  O.APBQ  is  harmonic  if 

C> Q 


Fig.  151 


a  transversal  MN  parallel  to  one  ray  OQ  is  bisected  by  the 
conjugate  OP. 

A  range  may  be  read  {AB,  PQ}  or  (APBQ),  and  a  pencil 
may  be  read  O(AB,  PQ)  or  O.APBQ.  Adopt  one  plan  and 
adhere  to  it,  or  the  boys  may  be  confused.  It  is  a  good  plan 
to  use  coloured  chalks  for  every  harmonically  divided  line 
on  the  blackboard,  and  always  of  the  same  colour.  Harmonic 
division  is  so  useful  that  its  immediate  recognition  is  desirable. 

The  Complete  Quadrilateral 

The  commonest  theorems  involving  harmonic  section 
concern  (1)  the  complete  quadrilateral,  (2)  pole  and  polar. 

Fig.  152  (i)  shows  a  tetragram  with  its  3  diagonals  (2  pro- 
duced to  meet)  which  are  indicated  by  heavy  lines.  Each  of 
the  3  diagonals  is  harmonically  divided  by  the  other  two.  Fig.  (ii) 
shows  a  tetrastigm  with  its  6  connectors  also  indicated  by 
heavy  lines,  and  with  lines  (faint)  joining  the  secondary 
points  in  pairs.  Each  of  the  6  connectors  is  harmonically  divided 
by  (1)  the  secondary  point  through  which  it  passes,  and  (2) 
the  line  joining  the  other  two  secondary  points.  If  we  superpose 
one  figure  on  the  other,  we  get  a  remarkable  series  of  harmonic 


316  CRAFTSMANSHIP  IN  MATHEMATICS 

pencils,  of  mutual  harmonic  intersections,  and  of  collinearities 
Actually,  of  course,  they  can  be  picked  out  in  fig.  152  (ii). 


(0 

Fig.  152 

If  the  complete  quadrilateral  is  approached  in  this  way, 
the  boy's  interest  and  curiosity  is  aroused.  He  is  greatly 
surprised  to  discover  that  a  simple  thing  like  a  quadrilateral 
has  so  many  remarkable  properties. 

But  the  important  thing  is  for  the  boy  to  realize  that  this 
general  quadrilateral  generalizes  the  theorems  of  all  particular 
quadrilaterals.  For  instance,  suppose  that  the  line  EF  in  the 
last  figure  is  removed  to  an  "  infinite  "  distance  from  C,  the 
4  points  C,  A,  D,  B,  become  the  vertices  of  a  parallelogram; 
and  since  R  is  the  harmonic  conjugate  of  the  point  in  which 
CD  intersects  EF  with  respect  to  the  points  C  and  D,  R 
becomes  the  middle  point  of  CD.  Thus  the  theorem  of 
the  complete  quadrilateral  is  a  generalization  of  the  theorem 
that  the  diagonals  of  a  parallelogram  bisect  each  other. 

Impress  on  the  boys  the  importance  of  the  use  of  a  general 
figure  in  their  geometrical  work,  and  the  fact  that  from  its 
properties  the  special  properties  of  a  particular  figure  may 
often  be  inferred. 

Pole  and  Polar 

The  dark  lines  in  the  figures  represent  harmonically 
divided  lines:  (i),  (ii),  every  chord  which  passes  through 
the  pole  P  is  cut  harmonically  by  the  polar;  (iii),  if  the  polar 
of  PX  passes  through  P2,  the  polar  of  P2  passes  through  Px. 
In  all  such  cases,  let  the  pole  and  the  corresponding  point 


MODERN  ADVANCED  GEOMETRY 


317 


on  the  polar  be  consistently  marked,  say,  by  P's.  Also  let  the 
circumference  be  cut  in  Q's.  A  consistent  system  of  naming 
all  harmonically  divided  lines  is  a  great  advantage. 


(0 


For  a  good   general   problem  on  harmonic   section,  see 
Scientific  Method,  pp.  387-9. 


Concurrency  and  Collinearity 

Many  theorems  involving  these  principles  form  excellent 
practical  problems  for  careful  work  in  junior  classes.  En- 
courage young  boys  to  bisect  the  angles  of  a  triangle,  to  bisect 
the  sides,  to  draw  perpendiculars  from  the  mid-points,  &c., 
and  to  make  discoveries  for  themselves.  Let  them  thus  obtain 
the  facts.  A  little  later,  the  simpler  theorems  and  their  proofs 
may  be  given;  e.g.  concurrent  lines  through  the  vertices 
of  a  triangle,  the  medians,  the  perpendiculars  to  the  opposite 
sides,  two  exterior  bisectors  and  the  internal  bisector  of  the 
three  angles.  A  little  later  still:  given  the  Menelaus  relation, 
prove  that  the  points  are  collinear;  given  the  Ceva  relation, 
prove  that  the  points  are  concurrent.  Pascal's  and  Brianchon's 
theorems  will,  of  course,  always  be  included.  Another  good 
type  of  theorem  is  this:  four  points  on  a  circle  and  the  tangents 
at  those  points  form,  respectively,  two  quadrilaterals  whose 
internal  diagonals  are  concurrent  and  form  a  harmonic  pencil, 
and  whose  external  diagonal  points  are  collinear  and  form 
a  harmonic  range.  The  principles  of  concurrency  and  col* 
linearity  are  so  important  that  they  cannot  be  too  strongly 


318  .         CRAFTSMANSHIP  IN  MATHEMATICS 

emphasized,  but  with  most  boys  facility  comes  only  after 
much  practice  with  varied  types  of  problems. 

Pascal's  theorem  suggests  the  study  of  the  hexastigm,  of 
which  that  theorem  is  the  simplest  property.  The  theorem  is 
usually  quoted,  "  The  opposite  sides  of  any  hexagon  in- 
scribed in  a  circle  intersect  in  3  collinear  points  ",  but  a 
more  precise  statement  is,  "  The  3  pairs  of  opposite  connectors 
of  a  hexastigm  inscribed  in  a  circle  intersect  in  3  collinear 
points  ".  Fully  expressed,  this  comes  to,  "  The  15  connectors 
of  a  hexastigm  inscribed  in  a  circle  intersect  in  45  points 
which  lie  3  by  3  on  60  lines  ". — I  have  seen  one  passable 
figure  prepared  by  a  boy;  he  was  looked  upon  as  the  fool  of 
his  Form,  though  he  was  extraordinarily  successful  in  the 
use  of  ruler  and  compasses.  Elaborate  drawing  of  this  kind 
is  largely  a  waste  of  time,  and,  after  all,  the  hexastigm  still 
remains  to  be  investigated  fully. 

The  Further  Study  of  the  Triangle  and  Circle 

There  are  numerous  theorems  on  the  triangle,  many  of 
them  simple,  many  useful,  many  beautiful.  For  instance, 
those  concerning  triangles  in  perspective,  pole  and  polar  with 
respect  to  a  triangle,  symmedian  points  of  a  triangle,  Brocard 
points  of  a  triangle. 

So  with  the  circle:  the  nine-point  circle,  escribed  circles, 
the  cosine  circle,  the  Lemoine  circle,  the  Brocard  circle. 

It  is  important  to  leave  on  the  boy's  mind  a  vivid  im- 
pression of  the  remarkable  properties  that  even  now  are 
frequently  being  discovered  concerning  the  triangle  and 
circle.  Boys  who,  when  they  leave  school,  know  no  more 
pure  geometry  than  that  contained  within  the  limits  of 
School  Certificate  requirements  are  certainly  not  likely  to 
devote  leisure  moments  to  a  systematic  playing  about  with 
circles  and  triangles,  in  the  hope  of  hitting  on  some  new  and 
perhaps  remarkable  property  yet  undiscovered. 


MODERN  ADVANCED  GEOMETRY  319 

Conic  Sections 

The  pure  geometry  and  the  algebraic  geometry  of  the 
cone  should  be  studied  side  by  side.  If  either  has  to  be 
sacrificed,  let  it  be  the  latter.  Algebraic  manipulation  is  all 
very  well,  but  the  cone  is  a  thing  which  occupies  space,  and 
when  its  spatial  relations  are  reduced  to  symbols,  these 
symbols  may  assume,  in  the  pupils'  minds,  an  importance 
which  is  not  justified,  and  the  geometry  proper  may  be 
overshadowed. 


CHAPTER  XXV 

Geometrical   Riders  and  their  Analysis 

In  those  schools  where  riders  are,  as  a  rule,  solved  readily, 
schools  where  boys  take  a  real  delight  in  attacking  new  ones, 
the  secret  of  success  seems  to  be  that  right  from  the  first 
every  new  theorem  and  every  new  problem  has  been  presented, 
not  as  a  thing  to  be  straight-away  learnt,  but  as  a  thing  to 
be  investigated  and  its  secret  discovered.  The  boys  do  not 
learn  a  new  theorem  or  problem  until  they  have  been  taught 
how  to  analyse  it,  and  to  discover  how  it  hangs  on  to  what 
has  gone  before. 

General  instructions  should  include  advice  as  to  the 
necessity  of  drawing  a  general  figure,  of  drawing  that  figure 
accurately,  and  of  setting  out  definitely  what  is  "  given  " 
and  what  is  to  be  proved.  We  append  a  few  instances  of 
problems  and  theorems  actually  solved  in  the  classroom, 
with  a  brief  summary  of  the  sort  of  arguments  used. 

1.  O  is  the  mid-point  of  a  straight  line  PQ,  and  X  is  a  point 
such  that  XP  =  XQ.  Prove  that  the  Z.XOP  is  a  right  angle. 


320  CRAFTSMANSHIP  IN  MATHEMATICS 

We  argue  in  this  way: 

(1)  What  facts  are  given? 

OP  =  OQ, 
XP  =  XQ. 

(2)  What  have  I  to  prove? 

That  ZXOP  is  a  right  angle. 

(3)  Since  I  have  to  prove  that  Z.XOP  is  a  rt. 

join  XO. 

x 


I  must 


(4)  How  have  I  been  able  to  prove  before,  that  an  L  is 

a  it.  Z? 

(i)  Sometimes  by  finding  it  to  be  one  of  the  two  eq. 
adj.  Z.s  making  a  str.  L. 

(ii)  Sometimes  by  finding  it  to  be  an  angle  in  a  semi- 
circle. 

(iii)  Sometimes  by  finding  it  to  be  at  the  intersection 
of  the  diag.  of  a  sq.  or  a  rhombus, 

(5)  The  first  of  these  looks  possible  here.    Are  the  adj. 

/.s   at  O   equal?     Yes,  if  the   two  As   XOP  and 
XOQ  are  congruent. 

(6)  Are  these  As  congruent?    Yes,  three  sides  in  the  one 

are  equal  to  three  sides  in  the  other,  as  marked. 

Now  I  know  how  to  write  out  the  proof,  in  the  ordinary 
way:  I  therefore  begin  again,  and  make  up  a  new  figure  as 
I  proceed. 


GEOMETRICAL  RIDERS  321 

Proof.     Join  XO.    In  the  As  XOP,  XOQ, 

OP  ==  OQ,        (given) 
XP  -  XQ,        (given) 
XO  is  common, 

.-.  A  XOP  =  A  XOQ;    (three  sides) 
i.e.  the  two  As  are  equal  in  all  respects. 
.-.  ^XOP-  ZXOQ, 
.- .  ^XOP  ==  90°.         (half  the  str.  £POQ) 

(which  was  to  be  proved) 

2.  ABCD  is  a  parallelogram;    E  is  the  mid-point  of  BC, 
and  AE  and  DC  produced  intersect  at  F.  Prove  that  AE  =  EF. 


Argument. 

(1)  What  facts  are  given? 

(i)  ABCD  is  a  /Z7m;    .• .  its  opp.  sides  are  ||. 
(ii)  BE  ==  EC  (by  constr.). 

(2)  What  have  I  to  prove? 

That  AE  -  EF. 

(3)  How  have  I  been  able  to  prove  before,  that  two  lines 

are  equal? 

(i)  Sometimes  by  finding  them  in  two  congruent  As. 
(ii)  Sometimes  by  finding  them  in  a  A  with  two  angles 

equal, 
(iii)  Sometimes  by  finding  them  to  be  the  opp.  sides 

of  a  £H7m. 

(B291)  22 


322 


CRAFTSMANSHIP  IN  MATHEMATICS 


(4)  Does  either  of  these  plans  seem  possible,  to  prove 

AE  eq.  to  EF? 

(5)  Yes,  the  first,  for  the  As  ABE  and  FCE  look  congruent. 

(6)  Are  they  congruent? 

(7)  Yes.    Two  Zs  and  a  side,  as  marked. 

Now  I  know  how  to  write  out  the  proof  in  the  ordinary  way 
Proof.    In  the  As  ABE,  FCE, 


BE  =  EC, 
ZAEB  =  ZFEC, 
ZABE  =  ZFCE, 
A  ABE  -A  FCE, 
.-.  AE-EF. 


(constr.) 

(vert.  opp.  Ls) 

(alt.  Ls;  BC  across  \\s  AE,  DF) 

(2  Z.s  and  a  side) 

(which  was  to  be  proved) 


3.  Draw  a  circle  of  |"  radius  to  touch  the  given  line  AE 
and  the  given  circle  CDE.  (The  given  line  must  not  be  more 
than  1"  from  the  given  circle.) 


(*> 


Fig.  156 


This  is  a  problem,  and  we  have  to  discover  the  method 
of  construction.  I  assume  the  problem  done,  and  I  make  a 
sketch  of  the  required  circle  in  position,  as  accurately  as 
possible  (fig.  156,  i).  I  examine  the  figure,  and  I  observe  that 
the  line  QR  to  the  pt.  of  contact  R  =  tf  and  is  ±  AB;  that 
HQ  =  HP  +  Y  and  passes  through  the  pt.  of  contact  P. 

Argument. 

(1)  Since  the  required   O  has  to  touch  the  line  AE>  its 


GEOMETRICAL  RIDERS  323 

centre  must  lie  somewhere  on  a  line  FG  ||  AB,  and 
Y  from  AB  (fig.  156,  ii). 

(2)  Since  the  required   O   has  to  touch  the    O    CDE,  its 

centre  must  lie  somewhere  on  the  circle  LMN 
having  the  same  centre  as  CDE  and  having  a  radius 
HQ  equal  to  radius  HP  +  ¥  (fig.  156,  ii). 

(3)  Since  the  centre  of  the  required    O  lies  both  on  the 

line  FG  and  on  the  circle  LMN,  it  must  be  at  a 
point  of  intersection  of  FG  and  LMN. 

Now  I  know  how  to  construct  the  circle. 
Construction. 

(1)  Draw  a  line  FG  ||  AB,  -|"  away  from  it. 

(2)  From  centre  H,  with  radius  equal  to  HP  +  i">  draw 
O  LMN. 

(3)  From  one  of  the  pts.  of  intersection  of  this  line  and 
circle,  say  Q,  as  centre,  draw  a  circle  RSP  of  |"  radius.    This 
is  the  required   O. 

Proof. 

(1)  The  circle  RSP  has  a  radius  of  |".    (constr.) 

(2)  The  circle  touches  AB  (in  R),  for  any  circle  of  \* 
radius  having  its  centre  on  the  line  FG  must  touch  AB,  a 
±   from  the  centre  Q  passing  through  the  pt.  of  contact. 
(constr.) 

(3)  The  circle  touches  the  given  circle  CDE  (in  P),  for 
any  circle  of  J"  radius  having  its  centre  on  the  circle  LMN 
must  touch  CDE,  the  line  joining  the  centres  passing  through 
the  point  of  contact,     (constr.) 

(4)  Therefore   the   circle   is   constructed   in   accordance 
with  the  given  conditions,    (which  was  to  be  done). 

4.  From  the  right  angle  of  a  right-angled  triangle,  one 
straight  line  is  drawn  to  bisect  the  hypotenuse,  and  a  second 
is  drawn  perpendicular  to  it.  Prove  that  they  contain  an  angle 


324  CRAFTSMANSHIP  IN  MATHEMATICS 

equal  to  the  difference  between  the  two  acute  angles  of  the 
triangle. 

A 


BED  c 

Fig.  157 

Given. 

(1)  a  rt.  L  at  A. 

(2)  rt.  Zs  at  E. 

(3)  DB  =  DC. 

(4)  DA  -  DB  —  DC.      (Since  a    O    will  go  round   the 

A  ABC  on  BC  as  diameter.) 

Further,  an  examination  of  the  figure  shows  that  in  the 
2  rt.  Zd  As  BAC,  BEA,  ZABE  is  common;  .'.  the  two  As 
are  equiangular.  This  fact  may  prove  useful. 

Required  to  prove:    ZEAD  —  (  ZABC  -  ZACB). 

Argument. — This  is  a  type  of  problem  in  which  we  may 
first  usefully  test  a  particular  case  by  assigning  to  some 
angle  a  number  of  degrees,  and  then  calculating  the 
number  in  some  or  all  of  the  other  angles.  For  instance, 
let  ZABC-  65°  (not  30°,  or  any  other  factor  of  360°,  lest 
a  fallacy  creep  into  our  argument). 

If  ZABC  =  65°,  ZACB  =  25°  (the  complement). 
If  ZACD  -  25°,  ZCAD  -  25°  (for  AD  -  DC). 
Also  ZBAE  =  25°  (equiangular  As,  as  above). 

Again,  if  ZABC  =  65°,  ZBAD  =  65°  (for  DB  =  DA), 
and  ZEAD  ==  ZBAD  -  ZBAE  -  65°  -  25°  =  40°. 

But   ZABC  -  ZACB  -  65°  -  25°  -  40°, 
/.    ZABC  -  ZACB  =  ZEAD. 


GEOMETRICAL  RIDERS  325 

Thus  the  theorem  is  true  in  this  particular  case.  We  are 
therefore  now  in  a  position  to  generalize  the  result  and  to 
set  out  the  proof  in  the  ordinary  way. 

Proof. 

(1)  DB  =  DA,  (given) 

.-.  ZDBA-  ZDAB. 

(2)  BAG  and  BEA  are  equiangular  rt.  Zd  As.     (given) 
.-.  ZACB-  ZEAB. 

(3)  .-.  ZDBA  -  L ACB  =  ZDAB  -  ZEAB  (from  1  and  2), 

-  ZDAE. 

(which  was  to  be  proved) 

5.  The  figure  shows  an  equilateral  triangle  ABC  within 
a  rhombus  ADEF,  a  side  of  the  former  being  equal  to  a  side 
of  the  latter.  Determine  the  magnitude  of  the  angles  of  the 
rhombus. 


Argument. 

From  an  examination  of  the  figure  we  know  the  following 
facts: 

(1)  Rhombus:  4  equal  sides;  opp.  sides  ||;  opp.  2Ls  equal. 

(2)  Equil.  A :   sides  equal;     Z.s  equal. 

(3)  As  ADB,  AFC  isosceles. 

(4)  A  EEC  isosceles  (by  symmetry). 


326  CRAFTSMANSHIP  IN  MATHEMATICS 

All  that  we  know  about   L  magnitudes  from  the  figure  are: 

(1)  Angle-sum  of  any   A  =  180°. 

(2)  Each  L  of  A  ABC  =  60°. 

We  have  therefore  to  try  to  express  the  Zs  of  the  rhombus  in 
terms  of  these  values. 

From  the  rhombus,   ZFAD  +   ZADB  =  2  rt.  Zs. 

Also,  ZABD  +   ZABE  =  2  rt.  Zs. 

But  ZADB  =  ZABD,  (isos.  A) 

.-.    ZFAD-  ZABE. 
Obviously,  therefore,  ZFAD-  ZFED-  ZABE-  /ACE 

Now  the  sum  of  the  last  3  of  these  Zs 

=  (sum  of  Zs  of  A  EEC)  +   ZABC  +   ZACB 
=  180°  +  120° 
-  300°. 

°  =  100°; 


.-.  ZADE  =  (180°  -  100°)  =  80°. 

The  estimate  may  be  set  out  formally  in  almost  the  same 
sequence. 

6.  Show  that  the  4=  straight  lines  bisecting  the  angles  of 
any  quadrilateral  form  a  cyclic  quadrilateral. 


Fig.  159 


GEOMETRICAL  RIDERS 


327 


Let  ABCD  be  the  given  quadrilateral,  and  let  the  bisectors 
of  the  Zs  form  the  quadrilateral  EFGH. 
Let  the  Zs  be  marked  as  shown. 

Given:  a  =  a';  j8  =  j8';  y  =  y';  8  ==  8'. 
Argument, 

If  a  circle  will  go  round  EFGH,  the  sum  of  any  2  opp. 

Zs  of  EFGH  —  2  rt.   Zs;  thus  a  +  T  =  2  rt.   Zs. 
//  a  +  r  =  2  rt.  Zs,  a  +  j8'  +  y  +  8'  =  2  rt.  Zs  since 

the  sum  of  all  the  As  of  the  2  As  ABE  and  CDG 

=  4  rt.  Zs. 
But  a  +  /J'  +  y  +  8'  we  &wo«;  are  equal  to  2  rt.  Zs,  for 

2a  +  2)3'  +  2y  +  28'  =  4    rt.  Zs    (the   4  Zs    of    the 

quadl.) 
Thus  we  have  found  the  key. 

Proof. 

The  sum  of  the  4  Zs  of  the  quadl.  ABCD  =  4  rt.  Zs. 

.• .  the  sum  of  the  halves,  a  +  ft'  +  y  +  8'  =  2  rt.  Zs. 
But  the  sum  of  all  the  Zs  of  As  ABE  and  CDE  =  4  rt.  Zs. 
.• .  a'  +  T'  =  2  rt.  Zs, 
.-.a   +  T  =  2  rt.  Zs, 
.-.  the  points  E,  F,  G,  H  are  concyclic. 

(which  was  to  be  proved) 

7.  Three  points  D,  E, 
and  F  in  the  sides  of  a  tri- 
angle ABC  are  joined  to 
form  a  second  triangle,  so 
that  any  two  sides  of  the 
latter  make  equal  angles  with 
that  side  of  the  former  at 
which  they  meet.  Show  that 
AD,  BE,  and  CF  are  at 
right  angles  to  BC,  CA,  Fig.  I6o 

and  AB,  respectively.    (You 
may   not    assume   properties    of   the   pedal    triangle.) 


328  CRAFTSMANSHIP  IN  MATHEMATICS 

Given:  a  =  a';  0  =  £';  y  =  y'. 
Required  to  prove:   AD  is  -L  to  BC,  &c. 
Argument.  —  Assume  that  AD  is  -L  to  BC. 
Then  •  .*  a  =  a',  S  =  S'. 
DF       FG 


= 
FG       GE' 

DA 


But  this  is  known,  since  each  ratio  = 


AG' 


LThis  is  easily  shown:    Produce  DE  to  H;    EA  is  the 
DF       DA 

bisector  of  the  external  ZGEH  of  the  A  GDE;   .-.=-=  =  —  . 

EG      AG 

Similarly,  by  producing  DF  to  K,  it  is  seen  that  -—  .  =    -_  . 

AG  J 


Thus  we  can  make  these  known  ratios  our  starting-points, 
and  set  out  the  proof  in  the  usual  way. 

Proof. 

Produce  DE  to  H  and  DF  to  K.  Then  EA  and  FA  are 
the  bisectors,  respectively,  of  ext.  ZGEH  of  A  GDE,  and 
of  ext.  -dGFK  of  AGDF. 

DE       DF  ,          ,        DA 
"  EG  =  FG'  f°r  CaCh  =  AG' 

DE  =  EG 
"  DF       FG' 
/.  DG  bisects  ZEDF, 
.'.  8  =  8', 
/.   8  +  a  =  8'  +  a', 
.-.  AD  is  ±  to  BC. 


GEOMETRICAL  RIDERS 


329 


Similarly,  we  may  show  that  BE  -L  AC,  and  CF  JL  AB. 

(which  was  to  be  proved) 

8.  Show  that  the  perpendicular  drawn  from  the  vertex 
of  a  regular  tetrahedron  to  the  opposite  face  is  3  times  that 
drawn  from  its  own  foot  to  any  of  the  other  faces. 

Let  ABCD  be  the  tetrahedron,  and  let  AE  be  the  -L  from 
the  vertex  A  to  the  opp.  face  BCD.  Then  E  is  the  centroid 
of  the  ABCD. 

Let  a  -L  EF  be  drawn  to  the  face  ACD;  F  will  meet  the 
median  AG. 


SECTION  THROUGH  TETRAHEDRON 
Fig.  162 


To  prove:  AE  =  3EF. 

Argument:   Consider  the  vertical  section  through  ABG. 

We  know  that,  since  E  is  the  centroid  of  BCD,  EG  =  £  BG, 
Is  there  an  analogous  relation  between  EF  and  AE? 

If  we  draw  BK   1  AG  in  the  face  ACD,  BK  must  be 
equal  to  AE. 

E F  _  EG  =  i 

BK    "  BG       3' 

/.    EF  =  JBK  =  |AE 


For  the  other  faces  similar  results  follow  from  symmetry. 


330  CRAFTSMANSHIP  IN  MATHEMATICS 

Proof. 

Let  a  1  BK  from  B  meet  the  median  AG  in  K;  BK  =  AE. 
EG  =  £BG.        (E  is  the  centroid  of  BCD) 
EF  _EG       , 
BK      BG       *' 

/.  EF  -  PK 

=  JAE.          (which  was  to  be  proved) 

If  a  rider  is  in  any  way  of  an  unusual  character,  pupils 
sometimes  have  difficulty  in  writing  out  a  proof  concisely. 
We  give  an  example  of  an  acceptable  proof  for  such  a  rider. 


Fig.  163 

In  a  given  triangle  ABC,  BD  is  taken  equal  to  one-fourth 
of  BC,  and  CE  equal  to  one-fourth  of  CA.  Show  that  the 
straight  line  drawn  from  C  through  the  intersection  F  of  BE 
and  AD  will  divide  the  base  into  two  parts  at  G  which  are 
in  the  ratio  9  to  1. 

ABEA-3  A  EEC, 

AFEA  =  3  AFEC, 

.'.    ABFA-=3  ABFC, 

=  12  ABFD, 
.'.  AF  =  12  FD, 
/.    AAFC=  12  ADFC 
=  36  ABFD 
=  9  ABFC. 


GEOMETRICAL  RIDERS  331 

Now  the  As  AFC,  BFC  are  on  the  same  base  FC.  Hence  the 
vertical  height  of  A  AFC  above  this  base  =  9  times  the 
vertical  height  of  A  BFC  above  this  base. 

.'.   A  AGC  —  9  ABGC,        (on  the  extended  base,  GC) 
AG  =  9  GB. 

Any  reasonable  examiner  would  accept  a  proof  given  in 
this  form  and  would  be  glad  to  be  saved  from  the  trouble  of 
reading  defensive  explanatory  matter. 

Books  on  geometry  to  consult: 

1.  Plane  Geometry,  2  vols.,  Carson  and  Smith. 

2.  Geometry,  Godfrey  and  Siddons. 

3.  Geometry,  Barnard  and  Child. 

4.  Elementary  Concepts  of  Algebra  and  Geometry,  Young. 

5.  Elementary  Geometry,  Fletcher. 

6.  Cours  de  Ge'ome'trie,  d'Ocagne  (Gauthier  Villars). 

7.  A  Course  of  Pure  Geometry,  Askwith. 

8.  Modern  Pure  Geometry,  Lachlan. 

9.  Sequel  to  Elementary  Geometry,  Russell. 

10.  Geometry  of  Projection,  Harrison  and  Baxandall. 

11.  Protective  Geometry,  Matthews. 

12.  An  Elementary  Treatise  on  Cross-Ratio  Geometry,  Milne. 

13.  Foundations  of  Geometry,  Hilbert. 

14.  The  Elements  of  Non-Euclidean  Geometry,  Sommerville. 

15.  Space  and  Geometry,  Mach. 

16.  Analytical  Conies,  Sommerville. 

17.  Curve  Tracing,  Frost  (new  edition).    An  old  and  faithful  friend. 

18.  Euclid,  3  vols.,  Heath.    The  work  on  the  subject. 


332  CRAFTSMANSHIP  IN  MATHEMATICS 

CHAPTER  XXVI 
Plane  Trigonometry 

Preliminary  Work 

The  pupils'  first  /acquaintance  with  the  .tangent,  sine, 
and  cosine  should  be  made  during  their  elementary  lessons 
in  geometry.  Boys  soon  (learn  that  the  symbols  for  the 
trigonometrical  ratios^  may  enter  into  formulae  which  can 
be  manipulated  algebraically;  and  since,  in  the  algebra 
course,  the  study  of  xn  and  ax  is  included,  it  is  difficult  to 
exclude  from  it  the  study  of  sin  x  and  tan  x.  Each  represents 
a  typical  kind  of  function.  To  each  corresponds  a  specific 
form  of  curve — its  own  particular  picture,  the  graphic  picture 
of  the  function.  Algebra  and  trigonometry  should  be  much 
more  closely  linked  together,  and  much  of  the  purely  formal 
side  of  trigonometry  might  with  advantage  be  sacrificed, 
and  greater  stress  be  laid  on  the  practical  and  functional 
aspects  of  the  subject.  The  needs  of  co-ordinate  geometry 
and  the  calculus,  of  mechanics  and  physics,  should  always 
be  borne  in  mind;  in  fact,  much  of  the  work  done  in  trigo- 
nometry might  be  directed  towards  these  subjects. 

The  notion  of  an  angle  as  a  rotating  line  should  be  given 
at  the  very  outset  of  geometry,  so  that,  when  in  trigonometry 
angles  greater  than  180°  are  discovered,  the  notion  will 
already  be  familiar.  The  angle  of  "  one  complete  rotation  ", 
and  its  subdivisions,  straight  angle,  right  angle,  and  degree, 
will,  of  course,  be  known,  and  pupils  should  be  able  to  draw 
freehand,  at  once,  to  a  fairly  close  approximation,  an  angle  of 
any  given  size,  the  30°,  45°,  and  60°  angles  being  quite 
familiar  from  the  half  equilateral  triangle  and  the  half  square. 

Co-ordinate  axes  and  the  four  quadrants  will  also  be 
familiar  from  previous  work  on  graphs;  so  will  directed 
algebraic  numbers.  Angles  of  elevation  and  depression  will 
already  have  been  measured  in  connexion  with  practical 


PLANE  TRIGONOMETRY 


333 


problems  in  geometry  and  mensuration.  Pythagoras  should 
be  at  the  pupils'  finger-ends;  so  should  the  fundamental 
idea  of  projection}  Similar  triangles  should  also  be  known, 
and  ratios  of  pairs  of  sides  should  be  equated  with  readiness. 
Unless  all  these  things  are  known,  really  known,  the  earlier 
work  in  trigonometry  is  much  hampered  by  time-consuming 
preliminaries. 

Do  not  scare  the  class  in  the  first  lesson  by  hurling  at 
their  heads  all  six  trigonometrical  ratios.  Only  the  tangent, 
sine,  and  cosine  need  be  studied  at  first,  and  these  one  at  a 
time,  each  as  a  natural  derivative  of  practical  problems  of 
some  kind. 

The  Tangent 

The  tangent  should  come  first.      Revise  a  few  simple 
geometry    problems    in    heights    and    dis- 
tances,   and    let    the    new    trigonometrical 
term  gradually  replace  the  geometrical  ratio 
which  the  boys  already  know. 

We  might  begin  in  this  way. 

Measure  the  height  of  the  school  flag-staff 
AB. 

Set  up  the  4'  high  theodolite  at  D,  at 
a  distance  of,  say,  25'  from  B,  and  measure 
the  angle  AEC  (==  58°).  Make  a  scale 
drawing.  By  scale,  AC  =  40'.  Hence  AB 
=  AC  +  CB  =  40'  +  4'  =  44'. 


Thus  the  ratio  -  ~  =  1-6. 

In  other  words,  when  the  angle  E  is  58°,  AC  =  1-6  EC. 
Now  look  at  a  series  of  right-angled  triangles  with  the  base 
angle  58°.  In  every  case  the  ratio  AC/CE  is  the  same,  since 
the  triangles  are  similar.  Thus  in  each  case  AC  =  1-6  EC. 
Hence,  whatever  the  length  of  EC,  we  can  find  the  length 
of  AC  by  multiplying  EC  by  1-6.  (Fig.  165.) 


334 


CRAFTSMANSHIP  IN  MATHEMATICS 


Thus  the  number   1-6  is  evidently  associated  with  the 
particular  angle  58°.    How?    It  measures  the  ratio  AC/CE, 

i.e.  the  P^Pendicular  of  the  right-angled  triangle  AEC.     If, 

base 

then,  we  make  a  note  of  this  value  1-6,  as  belonging  to  the 
particular  angle  58°,  we  are  likely  to  find  it  very  valuable 
when  dealing  with  right-angled  triangles  having  an  angle 


E  c      E  c         E  c 

Fig.  165 

of  58°;  if  we  know  the  base  we  have  merely  to  multiply  it 
by  1-6  to  obtain  the  perpendicular.* 

Obviously  every  angle,  not  merely  58°,  must  have  a  special 
value  of  this  kind.  We  may  take  a  series  of  right-angled  tri- 
angles, with  different  base  angles,  say  10°,  20°,  30°,  40°,  50°, 
60°,  70°,  80°,  measure  their  perpendiculars  and  bases  to  scale, 
calculate  their  ratios,  and  make  up  a  little  table  for  future  use. 

If  we  liked,  we  could  draw  these  triangles  independently, 
though  that  would  make  the  arithmetic  rather  tedious.  An 
easier  way  is  to  draw  a  base  of  exactly  1"  in  every  case;  then 
our  arithmetic  is  easy  (fig.  166).  (Any  number  instead  of  1 
would  do,  but  that  would  mean  a  little  more  arithmetic.) 


•58; 


EC      -36 


•  Do  not  mention  the  term  hypotenuse  at  all. 
dealt  with. 


Let  that  wait  until  the  sine 


PLANE  TRIGONOMETRY 


835 


Mathematicians  sometimes  make  the  perpendicular  a 
tangent  to  the  circle,  fig.  167  (they  always  remember  that 
an  angle  is  concerned  with  rotation):  and  for  convenience 

they  call  the  ratio  PerP^ndlcu_lar  the  tangent  of  the  angle. 

base 

Thus  they  say,  tangent  10°  ==  -18;   tangent  20°  =  -36;     and 
so  on.  They  generally  write  tan  for  tangent. 


Bi 


-rj- 
•84  ; 


Fig.  I 66 


Fig.  167 


But  remember  that  the  tangent  of  an  angle  is  just  a  number 
which  shows  how  many  times  the  perpendicular  is  as 

perp. 


long  as  the  base;   in  other  words,  it  is  the  ratio 


Since 


perp. 

^—*-  -  tan, 
base 


perp.  =  base  X  tan\ 


base 
hence  in  the 


triangle  ABC,  AC  =  BC  X  tan  35°, 
i.e.  the  tan  of  an  angle  is  the  mul- 
tiplier for  converting  the  base  into 
the  perpendicular.  (Fig.  168.) 

There  are  better  ways  of  finding 
these  values  than  by  merely  drawing 
to  scale;  in  fact,  values  to  7  places  of 
decimals  have  been  found,  the  work 
to  be  done  with  them  (by  surveyors, 
for  instance)  having  often  to  be  very  accurate.  Here  is  a  little 


Fig.  1 68 


336 


CRAFTSMANSHIP  IN  MATHEMATICS 


table  giving  the  values  of  the  tangents  of  10  angles,  to  4  places 
of  decimals. 


tan  10°= 
„  20°  = 
„  30°  = 
„  40°  = 


-1763 
-3640 
-5774 
-8391 


45°  =  1-0000 


tan50°=     1-1918 
„    60°  =     1-7321 

„    70°  =     2-7475 
„    80°  -     5-671 
„    89°  =  57-29 


373 


y 

/ 

/ 

c       XT 

y 

0 

/ 

-"for 

75- 

373 

__/ 

/ 

/ 

| 

^^* 

^ 

/ 

,     •-• 

^-  — 

^  ' 

^  ' 

0 

l( 

5*     2 

0°    3 

0'    4 

A 

0*    5 

0°    <* 

0°     7 

r  a 
75- 

0*    9C 

Fig.  169 

There  is  no  tangent  for  90°.  Can  you  see  why?  Can  you 
see  why  the  tan  of  89°  is  so  large?  look  at  fig.  167.  Can 
you  see  why  the  tan  of  89°  59'  59"  must  be  enormously 
large? 

You  will  remember  how,  when  we  had  graphed  a 
function  of  #,  we  were  able  to  obtain  other  values  by 
interpolation.  We  may  do  the  same  with  the  tan  graph; 
in  fig.  169,  plotted  from  the  above  table,  you  may  see 
that  the  tan  of  75°  is  about  3-73.  To  get  anything 
like  accurate  values,  we  should  have  to  have  a  very  large 
graph. 

We  give  one  or  two  easy  practical  exercises. 


PLANE  TRIGONOMETRY 

A  ladder  leaning  against  a  house  makes  an 
angle  of  20°  with  the  wall.  Its  foot  is  10'  away. 
How  high  up  the  house  does  it  reach? 

We  have  to  obtain  the  height  AC,  and  we 
therefore  require  to  know  the  tan  of  the  angle  B. 
Since  A  -  20°,  B  =  70°. 


=  tanB  =  tan  70° 
BC 

=  2-7475  (see  table  or  graph). 
AC  ^  BC  X  2-74:75 
---  10'  X  2-7475 
=  27-475'. 


Fig.  170 


Two  boys  are  on  opposite  sides  of  a  flag-staff  50'  high. 
Their  angles  of  elevation  of  the  top  of  the  staff  are  20°  and 
30°,  respectively.  How  far  are  they  apart? 


Fig.  171 

Given,  length  of  AB;  Required,  length  of  BC  and  BD. 
Since  the  angles  at  C  and  D  are  given,  we  may  mark  in  the 
angles  at  A. 

Distance  of  boys  apart  —  CD 

=  CB  +  BD 

=  AB  tan  70°  +  AB  tan  60° 
=  50(2-7475  +  1-7321) 
-  223-98  (feet). 

Give  ample  practice  in  easy  examples  of  this  kind  until  the 
boys  are  thoroughly  familiar  with  the  fact  that  the  tan  is 
just  a  multiplier,  sometimes  less  than  1,  sometimes  greater, 

(B291)  23 


338 


CRAFTSMANSHIP  IN  MATHEMATICS 


for  calculating  the  length  of  the  base  from  the  per- 
pendicular. Vary  the  exercises,  so  that  the  base  is  not 
always  a  horizontal. 


The  Sine 

To  beginners,  navigation  problems  for  introducing  the 
sine  seem  to  be  a  little  difficult,  and  may  best  be  taken  a  little 
later.  Here  is  a  suitable  first  problem.  A  straight  level  road 
AB,  20  miles  long,  makes  an  angle  of  37°  with  the  west-east 
direction  AC.  How  much  farther  north  is  B  than  A? 

In  the  figure  we  have  to  find  the  length  BC.  It  is  easy  to 
find  this  length  from  a  scale  drawing:  BC  =12  miles,  i.e. 
B  is  12  miles  north  of  C. 


Fig.  172 


G 

Fig.  173 


BC 
BA* 


Now  examine  the  ratio  ^.    As  long  as  the  angle  A  in 


a  right-angled  triangle  remains  37°,  the  ratio  must  always 
be  the  same,  no  matter  what  the  length  of  the  sides,  e.g. 

BC       DE       FG     T-    k  f  r    ,. 

p  .  •    If  then  we  know  the  value  of  this  ratio 


for  one  triangle,  we  know  it  for  all  similar  triangles;  its 
value  is  ^j  °r  *6-  Thus,  if  AD  =  14,  DE  =  -6  of  14  =  -84; 
and  so  on. 

This  new  ratio  is  Perpendicular  and  is  called  sine.     It 

hypotenuse 

is  a  mere  number,  and  represents  how  many  times  the  per- 
pendicular is  as  long  as  the  hypotenuse.  We  ought  really 


PLANE  TRIGONOMETRY 


339 


to  say,  represents  what  fraction  the  perpendicular  is  of  the 
hypotenuse,  since  the  value  is  always  less  than  1.  Thus 
sine  37°  =  *6  (we  generally  write  sine,  sin,  though  we  pro- 
nounce "  sin  "  as  "  sine  "). 

Just  as  with  the  tangents,  so  with  the  sines:  we  might 
draw  a  series  of  right-angled  triangles  with  base  angles 
successively  equal  to,  say,  10°,  20°,  30°,  &c.,  and  so  construct 
a  table.  When  we  constructed  fig.  166  for  the  tangents,  we 
made  a  triangle  with  a  base  of  1  unit,  because  we  wanted  to 


Fig.  174 

make  the  arithmetic  easy,  and  then  the  base  was  the  de- 
nominator of  the  ratio.  In  the  case  of  the  sine,  we  will  also 
make  the  denominator  of  the  ratio  unity,  i.e.  we  must  now 
make  the  hypotenuse  unity.  Here  is  a  plan  for  doing  this. — 
With  O  as  centre,  and  unit  radius,  draw  a  circle.  With  the 
protractor,  mark  in  the  angles  10°,  20°,  30°,  &c.;  each  radius 
OB,  OC,  &c.,  is  equal  to  unity.  From  the  ends  B,  C,  D,  &c., 
of  these  radii,  drop  perpendiculars  to  the  base,  BG,  CN,  DK, 
&c.,  and  measure  them.  Since  OA  =  OB  =  OC  (&c.)  =  T, 
the  perpendiculars  will  be  fractions  of  1".  Now  we  may 

*     .  BG       -17  .   „  n      CN 

obtain  the  sines:    smlO   =  ~— -  ==  —  =  -17;    sm20   =  — r 

•34 
=  —  =  -34;  sin 30°  =  '50,  &c.    By  careful  measurement,  we 


340 


CRAFTSMANSHIP  IN  MATHEMATICS 


may  obtain  sines  to  2  decimal  places.     Here  is  a  little  table 
to  4  places. 


sin  10°  =  • 
„  20°  =  - 
„  30°  =  • 
„  40°  =  - 
„  45°  =  • 

(, 

1736 
3420 
5000 
6428 
7071 

si 
i 

n50 

»  60C 
,70f 
,  80C 
,  90C 

D    
5     
)     
)     
)    ' 

•766 
•866 
•939 
•984 
L-OOOi 

«o 

, 

^  

—-—'"'" 

9 

Sr> 

55°- 

1  • 

82 

/ 

S 

.0 

/ 

^ 

f 

, 

/ 

^iKiee.      «^? 

/ 

/ 

/ 

.X 

, 

/ 

/ 

• 

y 

•1 

/ 

10"     20*    30*    4CT    50*    60*    70*     80*    9O* 
65 

ANGLES 

Fig.  175 

By  drawing  the  sine  graph,  we  may  obtain  the  sine  of  any 
A     other  angle  up   to   90°,  by  inter- 
polation;  e.g.  sin  55°  is  about  -82. 
Remember  that  the  «W£  of  an 
angle   is  just  a  number.     Since 
perpendicular  _  ^      .  _ 

hypotenuse 

dicular  =  hypotenuse  X  sine.  Hence 
Fig.  .76  "    inthetriangleABC,AC=ABsin35°, 


PLANE  TRIGONOMETRY 


341 


i.e.  the  sine  of  an  angle  is  the  multiplier  for  converting  the 
hypotenuse  into  the  perpendicular.  In  this  case  the  multiplier 
happens  to  be  always  a  fraction. 

Here  are  one  or  two  easy  typical  problems: 

A  ladder  30'  long  stands  against  a  vertical  wall.  It  makes 
an  angle  of  70°  with  the  ground.  What  is  the  height  above 
the  ground  of  the  top  of  the  ladder?  (Fig.  177.) 

Given,  AB  =  30';    ZABC  =  70°.     Required  AC. 

AC 

—  =  sin  70°  =  -94  (from  table  or  graph), 
A.  ±5 

/.  AC  =  AB  x  -94  =  30'  x  -94  =  28-2'. 


A  railway  slopes  at  an  angle  of  10° 
for  a  distance  of  1000  yards.  What  is  the 
difference  in  level  of  its  two  ends?  (Fig. 
178.) 


Fig.  178 


Given,  AB  =  1000  yards;    Z.ABC  =  10°.     Required  AC. 
AC 


AB 


=  sinlO0  =  -1736. 


/.  AC  =  AB  X  -1736  =  1000  yd.  X  -1736  =  173-6  yd. 


The  Cosine 

Projection  problems  form  a  suitable  beginning. — AB 
represents  a  sloping  road  500  yd.  long.  A  surveyor  finds  that 
it  makes  an  angle  of  30°  with  the  horizontal  What  is  the 


342 


CRAFTSMANSHIP  IN  MATHEMATICS 


projected  length  on  a  horizontal  line,  such  as  would  be  shown 
on  an  ordnance  map? 

The  projection  of  a  line  AB  on  another  line  MN  is  the 
distance  between  two  perpendiculars  drawn  to  MN  from  the 


M 


Fig.  179 


N 


ends  of  AB.  If  MN  passes  through  A,  one  end  of  the  road, 
only  one  perpendicular  (BC)  is  necessary.  The  projection  is 
then  AC. 

Given,  AB  =  500  yd.;     ZBAC  =  30°.     To  find  AC. 

From  a  scale  drawing  we  find  that  AC  —  433  yd. 

AC 

Now  examine  the  ratio As  long  as  the  angle  A  in  a 

AJJ 

right-angled  triangle  remains  30°,  the  ratio  must  always  be 

B 


A  G       E       C 

Fig.  1 80 

the  same,  no  matter  what  the  length  of  the  sides,  e.g.  —  =  -- 
A  /-*  AJb>      AD 

=  pgr,  for  the  triangles  ABC,  ADE,  AFG  are  all  similar.    If 

then  we  know  the  value  of  this  ratio  for  one  triangle,  we  know 

433 
it  for  all  similar  triangles.     Its  value  is  —  or  -866.     This 

base  50° 

new  ratio,  ^      tenuse>  is  called  the  cosine  (generally  written 


PLANE  TRIGONOMETRY 
base 


343 


cos).    It  is  a  mere  number.    Since  f =  cos,  base  =  hypo- 

hyp. 

tem/5*  X  cos.  Hence,  in  the  triangle  ABC,  AC  =  AB  cos  30°, 
i.e.  the  cosine  of  an  angle  is  the  multiplier  for  converting 
the  hypotenuse  into  the  base.  In  this  case,  again,  the 
multiplier  always  happens  to  be  a  fraction. 

Nf 


Just  as  with  the  tangent  and  sine,  so  with  the  cosine: 
we  may  draw  a  series  of  right-angled  triangles  with  base 
angles  successively  equal  to  say  10°,  20°,  30°,  &c.,  measure 
them  up,  and  so  construct  a  table.  And  as  in  the  case  of  the 
sine,  we  will  so  construct  our  triangles  that  the  length  of  the 
hypotenuse  is  always  unity. 


344  CRAFTSMANSHIP  IN  MATHEMATICS 

Here  is  a  little  table  of  cosines,  to  4  places  of  decimals: 


cos  10°  =  -9848 
„  20°  =  -9397 
„  30°  =  -8660 
„  40°  =  -7660 
„  45°  =  -7071 


cos  50°  =  -6428 
„  60°  =  -5000 
n  70°  =  -3420 
M  80°  =  -1736 
„  90°  =  0 


By  drawing  a  cosine  graph  from  the  above  values,  we  can, 
by  interpolation,  obtain  the  value  of  any  other  angle  up  to 
90°,  e.g.  cos 35°  =  -82  (approx.). 


COSINES    '- 


•Q 

"  **^ 

"\ 

^ 

".A" 

SPS, 

_3S  = 

_-Q2 

\ 

O 
.7 

' 

\ 

./• 

\ 

\ 

\ 

•4 

\ 

.X 

\ 

, 

•2 

\ 

\ 

\ 

l( 

J      Z 

0     c5 

0     4 

0     £ 

0     G 

0     7 

0     « 

kO    9( 

55" 

DEGREES 

Fig.  182 

Compare  the  sine  and  cosine  graphs.  Each  is  an  exact 
looking-glass  reflection  of  the  other.  Now  look  at  che  two 
tables  of  sines  and  cosines.  Each  is  the  other  turned  upside 
down.  Evidently  there  is  a  curious  connexion  between  sines 
and  cosines. 

It  is  easy  to  draw  both  sine  and  cosine  curves  by  means 


PLANE  TRIGONOMETRY 


345 


of  intersecting  points  made  by  (1)  parallels  from  an  angle- 
divided  quadrant,  and  (2)  perpendiculars  from  the  corre- 
spondingly divided  abscissa.  Note  how  the  two  curves 
together  form  a  symmetrical  figure,  and  how  they  cut  in 
one  point.  What  do  you  infer  about  this  point  common 


\ 


0  O      10      20     30"  40*.  50"   60*  70*   60*  90 

4* 
Fig.  183 

to  the  two  curves?  There  is  evidently  some  angle  the  sine 
and  cosine  of  which  have  the  same  value.  Look  at  the  two 
tables. 

Easy  cosine  problems. — (1)  The  legs  of  a  pair  of  compasses 
are  5"  long.  Find  the  distance  between  the  points  when  the 
legs  are  opened  to  an  angle  of  80°. 


C 

Fig.  184 


Given:  AB  =  AD  =  5";  /.BAD  =  80°.    If  AC  is  the  bi- 
sector of  /.BAD,   ^BAC  =  40°;    hence   /.ABC  =  50°. 


346  CRAFTSMANSHIP  IN  MATHEMATICS 

Required:  length  of  BD  (=  2BC). 

•Dp 

±r  =  cos  50°;  /.  BC  =  AB  cos  50° 

=  5"  X  -64, 
.'.  BD  =  10"  x  -64  ==  6-4". 

(2)  C  is  any  point  in  the  line  XY.  CA  and  CB  are  drawn 
on  the  same  side  of  XY  so  that  CA  =  4",  CB  =  5",  LXCA 
=  40°,  L  YCB  =  60°.  Find  the  projection  of  ACE  on  XY. 


^  M  C  N  T 

Fig.  185 

Drop  perpendiculars  AM,  BN,  on  XY.    Then  the  projec- 
tion of  ACB  on  XY  is  MN.    Required:   the  length  of  MN. 

MN  =  CM  +  CN 

=  AC  cos  40°  +  BC  cos  60° 
=  (4"  X  -77)  +  (5"  X  -50) 
=  5-58". 

Now  give  the  boys  the  same  two  problems  again,  making 
them  use  the  sine  instead  of  the  cosine.  Hence  give  them 
the  first  notion  that  the  sine  and  cosine  are  so  closely  related 
that  one  may  sometimes  be  used  instead  of  the  other.  Make 
them  remember  this: 

If  the  hypotenuse  is  given, 

(1)  use  the  sine  to  find  the  perpendicular; 

(2)  use  the  cosine  to  find  the  base. 


PLANE  TRIGONOMETRY 


347 


The  sin,  cos,  and  tan:    Simple  Inter-relations 

Introduce  the  notation  a,  by  and  c  to  represent  the  number 
of  units  of  length  in  the  sides  opposite  the  angles  corre- 


spondingly named.     Also  show  that  since  Z.A  +  Z.B  =  90° 
A  =  90°  —  B,  and  B  =  90°  —  A.     Now  tabulate: 

-  =  tanB,  or  b  =  a  tanB;     -  =  tanA,  or  a  —  b  tanA. 
a  b 

-  =  sinB,  or  b  =  c  sinB;     -  =  sinA,  or  a  =  c  sinA. 
c  c 

_  =  cosB,  or  a  =  c  cosB;     -  =  cosA,  or  b  =  c  cosA. 
c  c 

Hence, 

(1)  Since  ?  =  tanA,  and  -  =  tanB,    /.  tanA  = 
b  a 


(2)  Since  -  =  sinA  =  cosB, 


(3)  Since  -  =  cos  A  =  sinB, 
c 


w      l=b' 

c 

(5)  Similarly, 


tanB 
sinA  =  cosB. 

cosA  =  sinB. 
sinA 


cos  A 
sinB 


=  tanA. 


tanB* 


348 


CRAFTSMANSHIP  IN  MATHEMATICS 


(6)  Since  B  =  90°  —  A,  and  sin  A  =-  cosB, 

.*.   sinA  =  cos(90°  —  A). 

(7)  Since  B  =  90°  —  A,  and  cos  A  =  sinB, 

/.   cos  A  =  sin  (90°  —  A). 

All  these  relations  must  be  carefully  committed  to  memory.* 
Note  that  the  last  two  may  be  summed  up  in  this  way:  the 
sine  of  an  angle  is  the  cosine  of  its  complement.  Explain  the 
significance  of  co-  in  cosine. 

Some  teachers  prefer  the  words  opposite  and  adjacent 
instead  ^perpendicular  and  base,  but  experience  suggests  that 
for  beginners  the  latter  terms  are  preferable.  The  main 
thing  is  to  adopt  one  form  of  words  and  stick  to  it. 

The  secant,  cosecant,  and  cotangent.  —  These  should  be 
remembered  as  the  reciprocals  of  the  cos,  sin,  and  tan, 
respectively.  Give  easy  examples  to  show  the  appropriateness 
of  the  forms  beginning  with  co. 

The  ratios  of  common  angles. — The  sin,  cos,  and  tan  of 
the  common  angles  30°,  45°,  and  60°  should  be  memorized 


Fig.  187 

as  soon  as  the  nature  of  the  three  functions  is  understood. 
Teach  the  boys  to  visualize  the  half  square  and  the  half 
equilateral  triangle — the  obvious  aids  to  memory. 

*  Dp  not  despise  some  simple  form  of  mnemonics  when,  with  beginners,  con- 
fusion is  almost  inevitable,  as  in  the  case  of  the  three  trigonometrical  functions;  e.g. 
remember 


(1)  Tan  = 

(2)  Sin   == 

(3)  Cos  = 


by  the  words  Tanned  Post  Boy, 
by  the  words  Sign,  Please,  Henry, 
by  the  words  Costly  Black  Hat, 


or  some  other  form  of  catchy  words, 


PLANE  TRIGONOMETRY 


349 


30° 

45° 

60° 

1 

V3 

sin 

4 

V2 

T~ 

V3 

1 

cos 

i 

1 

tan 

1 

V3 

V3 

A  little  later,  the  table  should  be  extended  to  0°  and  to 
90°,  and  eventually  to  180°.  When  discussing  the  0°  and 
90°  values,  draw  a  series  of  right-angled  triangles,  beginning 
with  a  very  small  acute  angle  A  and  very  nearly  0°,  and  ending 
with  an  angle  A  very  nearly  90°.  A  discussion  of  just  one 
general  figure,  without  reference  to  the  actual  values  of 
particular  cases,  is,  with  beginners,  almost  profitless.  Do 
not  say  that  the  tan  of  90°  is  "  infinity  ",  a  term  which  is 
beyond  the  comprehension  of  beginners.  Adopt  some  such 
non-committal  form  of  words  as  "  immeasurably  great  ". 


* 


Fig.  188 


The  ratio  for  15°  is  easily  obtained  from  this  figure, 
derived  from  fig.  187.  ABC  is  an  isosceles  rt.  Z.d  A,  sides 
1,  1,  2,  angles  45°,  45°,  90°.  ABD  is  a  half  equil.  A,  sides 
1,  2,  v%  angles  30°,  60°,  90°.  Thus  ^CAD  =  15°,  and 


350  CRAFTSMANSHIP  IN  MATHEMATICS 

CD  =  (-\/3  —  1).     From  C,  drop  a  perpendicular  on  AD. 

Since    CED   is   a   half   equil.    A,    CE  =  £CD  =  ^3  ~  1. 

pp       A/3  _  1 
Hence  sin  15°  —  —  -'  —  ~  ---    From  this  the  other  ratios 


of  15°  are  easily  found,  and  then  those  of  75°. 

For  18°,  fig.  113fl  is  the  key.  The  small  angles  of  a  regular 
pentagram  are  36°,  and  hence  the  sine  of  half  the  angle  is 

—  —  -  -  .     Let  the  boys  work  this  out  for  themselves;    it  is 
4 

a  good  exercise;  the  other  ratios  may  be  derived  arithmetically, 
but  the  first  (the  sine)  must  be  established  geometrically. 
The  derivation  for  multiples  of  18°  (36°,  54°,  72°)  is  suitable 
work  a  year  later. 

The    following    identities    may    readily    be    established 
geometrically. 

1.  Sin2A  +  cos2A  =  1.  —  This  is  seen  from  a  figure  to 
be  a  direct  application  of  Pythagoras.     Let  the  derivatives 
also  be  noted:   sin  A  =  A/1  —  cos2A,  cos  A  =  A/1  —  sin2A. 

2.  1  +  tan2A  =  sec2A.  —  Here  a  hint  is  necessary  to  the 
boys  to  work  "  backwards  ".    We  have  to  prove: 

J?  j        BC2       AB2 


i.e. 


AC2      AC2' 
AC2  +  BC2       AB2 


Fig.  is9  AC2  AC2' 


The  boys  now  observe  that  the  numerators  form  the  simple 
Pythagoras  relation.    Hence  they  write  out: 

AC2  +  BC2  =  AB2,  (Pythag.) 

.    AC2       BC2       AB2  .     , 

' 


/.     1  +  tan2A  =  sec2A.  Q.E.D. 

The  obvious  derivatives  should  follow.  —  Give  several  easy 


PLANE  TRIGONOMETRY  351 

examples  to  verify  the  rule  that  if  any  one  trigonometrical 
ratio  of  an  angle  be  given,  the  other  ratios  may  all  be  cal- 
culated without  reference  to  tables.  But  all  fundamental 
relations  must  be  established  geometrically.  Geometry  must 
take  precedence  over  algebra. 

Heights  and  Distances 

It  is  surprising  what  a  great  variety  of  problems,  in 
three  as  well  as  in  two  dimensions,  may  be  solved  by  means 
of  the  small  amount  of  trigonometry  already  touched  upon. 
Give  plenty  of  such  problems  until  the  sin,  cos,  and  tan 
are  as  familiar  as  the  multiplication  table,  are,  indeed,  a 
part  of  the  multiplication  table.  Insist  all  along  that  every 
problem  on  heights  and  distances  is  really  a  geometry  problem 
with  an  arithmetical  tail,  but  that  the  arithmetic  is  made 
easy  for  us  because  all  the  necessary  multiplication  sums 
have  been  worked  out  and  the  answers  put  into  a  book  of 
tables,  the  multipliers  having  been  given  the  rather  fanciful 
names  of  sin,  cos,  tan,  &c.  In  every  problem  we  are  con- 
cerned with  a  triangle;  the  length  of  one  side  is  always 
given,  and  the  multipliers  in  the  book  of  tables  enable  us  to 
find  the  other  sides;  to  find  the  multipliers,  we  have  to  know 
the  angles  of  the  triangle. — Four-figure  tables  of  natural 
sines,  cosines,  and  tangents,  for  whole  degrees  only,  are 
enough  for  beginners.  Let  logs  wait.  Let  the  problems  be 
easy  and  varied.  Three-dimen- 
sional problems  may  be  in- 
cluded quite  soon,  though  at 
least  a  little  solid  geometry 

should    have   been   done   pre-     W 

viously. 

When  setting  problems  in- 
volving "  bearings  ",  avoid,  as  ,. 
a  rule,  the  old  terms  "  north-                       Fig.  190 
west  ",  "  south-east  ",  &c.,  and 
adopt  the  surveyor's  plan,  always  placing  N.  or  S.  first, 


352  CRAFTSMANSHIP  IN  MATHEMATICS 

then  so  many  degrees  W.  or  E.,  thus  N.  30°  W.,  S.  60°  E. 
the  angle  always  being  measured  from  the  N. — S.  line. 

The  drawing  of  figures  for  heights  and  distances.  If  a 
figure  lies  wholly  in  a  horizontal  plane,  there  is  seldom  much 
difficulty,  especially  if  drawing  to  scale  has  been  properly 
taught  in  the  Junior  Forms.  Figures  in  a  vertical  plane  are 
also  readily  drawn,  though  the  angles  of  elevation  and  de- 
pression are  sometimes  confused  by  boys  whose  early  practical 
geometry  has  not  been  properly  taught. 

Consider  this  old  problem: 
From  a  point  P  in  a  horizontal 
plane,  an  observer  notes  that  a 
distant  inaccessible  tower  subtends 
an  angle  of  30°.  He  walks  to 
Q,  a  distance  of  100/J.,  towards 
the  tower,  and  finds  that  the  tower 
then  subtends  50°.  Find  the  height 
Fig.  191  of  the  tower  and  the  man's  dis- 

tance from  it. 

Explain  how  easy  it  is  to  work  with  tangents,  as  the  figure 
readily  shows. 

(1)  RS  =  PS  tan 30°,  i.e.  x  =  (y  +  100)  tan 30°. 

(2)  RS  =  QS  tan50°,  i.e.  x=y  tan50°, 

.'.  (y  +  100)  tan  30°  -  y  tan  50°. 

Hence  y  can  be  found,  then  x  by  substitution.  The  long 
succession  of  statements  in  some  of  the  textbooks  is  un- 
necessary and  merely  serves  to  bewilder  the  boys. 

The  problem  is,  of  course,  easy  enough.  It  is  only  when 
the  measured  distance  PQ  is  not  in  the  same  plane  as  PRS, 
i.e.  is  not  directly  towards  the  tower,  that  the  boys  are  baffled, 
because  of  the  difficulty  of  drawing  a  suitable  figure  in  3 
dimensions. 

We  will  deal  with  the  three-dimensional  figure  difficulty 
in  one  or  two  problems: 

A  wall  12  ft.  high  runs  east  and  west.     The  sun  bears 


PLANE  TRIGONOMETRY 


353 


Calculate  the  breadth  of  the 


S.  60°  W.  at  an  elevation  of  32°. 
shadow  of  the  wall  on  the  ground. 

This  is  taken  from  one  of  the  best  of  our  textbooks, 
is,  of  course,  very  simple,  yet  S 

I  have  given  it  to  several  lots  of 
boys,  and  the  necessary  figures 
have  nearly  always  puzzled  them. 
Had  the  sun  been  directly  south, 
a  stick  placed  vertically  at  O 
would  have  had  its  shadow  cast 
on  the  ground  in  the  direction 

ON  (fig.  192).    But  as  the  sun  was  S.  60°  W.,  the  stick  at  O 
would  have  had  its  shadow  cast  on  the  ground  in  the  direc- 


It 


W 


N 

Fig.  192 


•  W 


N 
Fig.  193 


tion  OQlt  so  that  the  shadow  makes  an  angle  of  30° 
the  vertical  plane  (wall)  in  EW  (fig.  193). 

SUN 


with 


O 
Fig.  194 


But  as  the  sun  has  an  elevation  of  32°,  the  length  of  the 
shadow  of  the  stick  RO  would  be  Q2O,  Q2  being  the  far  end 
of  the  shadow  on  the  ground  (fig.  194).  (During  the  day 


(B291) 


354 


CRAFTSMANSHIP  IN  MATHEMATICS 


this  shadow  would  occupy  a  succession  of  positions,  just  as 
if  it  were  pivoted  on  the  stick,  following  the  sun  round.) 

We  have  to  consider  these  two  things,  the  direction  and 
the  length  of  the  shadow  in  a  three-dimensional  figure.  Let 
ABCD  be  the  wall  running  east- west;  it  may  be  looked 
upon  as  a  close  set  of  palings,  with  one  paling  RO  taking 
the  place  of  the  stick.  Of  course  the  shadow  of  the  whole 
wall  will  be  cast,  but  we  will  first  consider  the  shadow  of 
RO  only.  As  the  sun  is  at  32°,  the  shadow  must  be  cast 

rS\JH 


Fig.  195 


somewhere  on  the  ground  as  a  length  OQ.  That  "  somewhere  " 
is  given  us  by  the  sun's  position  (irrespective  of  its  height) 
at  a  particular  time  in  the  day,  viz.  S.  60°  W.,  i.e.  OQ  will 
make  30°  with  the  east-west  wall,  or  ZBOQ  =  30°. 

Now  all  the  palings  will  cast  shadows  parallel  to  OQ, 
and  thus  we  shall  have  a  belt  of  shadow,  on  the  ground 
BMNC,  of  a  breadth  equal  to  the  perpendicular  QP  to  the 
wall.  Thus  we  have  to  find  the  length  of  PQ. 

To  find  the  length  of  PQ  we  may  solve  the  APQO  in  the 
H.P.  In  the  ARQO  (vertical  plane),  RO  =  12';  QO  =  RQ 
cot32°  =  RO  tan58°  =  12'  x  1-6  =  19-2'.  In  the  APQO, 
PQ  =  QO  sin30°  =  19-2'  X  -5  =  9-6'. 


PLANE  TRIGONOMETRY 


355 


For  beginners  a  model  is  far  better  than  a  sketch;  then 
the  angles  do  not  mislead.  Even  a  book  held  upright  on  the 
desk  to  represent  a  vertical  plane,  and  then  a  pencil  placed 
in  position  to  represent  a  line  in  an  oblique  plane,  will  help 
the  eye  greatly.  But  some  long  hat-pins  stuck  vertically 
into  a  board,  with  pieces  of  cotton  tied  round  under  the 
heads  (a  snick  made  with  the  laboratory  file  will  help  to 
secure  the  cotton),  stretched  and  held  fast  by  a  twist  under 
the  head  of  a  drawing-pin,  will  enable  the  boys  to  make  in 
a  minute  or  two  a  model  of  almost  any  figure  that  may  be 
required. 

Here  is  another  problem  and  the  provided  figure  from  the 
same  excellent  textbook.  The  figure  has  puzzled  several 
lots  of  boys. 

"  A  hillside  is  a  plane  sloping  at  27°  to  the  horizontal.  A 
straight  track  runs  up  the  hill  at  an  angle  of  34°  with  a  line 
of  greatest  slope.  What  angle  does  the  track  make  with  the 
horizontal?" 

"  AB  is  the  line  of  intersection  of  the  hillside  and  a 
horizontal  plane  ABC.  AF,  BE  are  lines  of  greatest  slope 


Fig.  196 


meeting  a  horizontal  at  F,  E.  Let  the  track  AD  cut  EF  at 
D.  Draw  DN,  EC  perpendicular  to  the  H.P.,  ABC.  Then 
AN  is  the  projection  of  AD  on  ABC.  It  is  required  to 
find  ZDAN  =  0,  say."  Then  follows  the  solution,  simple 
enough,  of  course,  from  considerations  of  the  3  it.  Zd  As 
ECB,  AND,  AFD,  the  first  two  in  V.P.s,  the  last  in  an  oblique 


356 


CRAFTSMANSHIP  IN  MATHEMATICS 


plane.  A  model  with  3  hat-pins  at  FM,  DN,  and  EC,  and 
drawing-pins  at  A,  B,  C,  M,  N,  and  connecting  threads, 

would  make  the  whole 
thing  clear  at  once. 
Otherwise  a  few  shading 
lines  might  be  added,  as 
in  fig.  197;  the  3  planes 
are  then  shown  clearly. 

Here  is  a  simple 
problem  from  another 
book,  the  figure  for  which 
has  often  given  beginners 

trouble. —  The  extremity  of  the  shadow  of  a  flag-staff  FG,  6' 
high,  standing  on  the  top  of  a  square  pyramid  34'  high,  just 
reaches  the  side  of  the  base  and  is  distant  56'  and  8'  respectively 
from  the  extremities  of  that  side.  Find  the  sun's  altitude. 


Fig.  197 


Fig.  198 

FK  =  FG  +  GK  =  6'  +  34'  =  40'.  We  have  to  find 
the  Z.FMK  in  the  rt.  Zd  A  FMK.  In  this  A  we  know 
FK;  and  we  can  find  MK  by  Pythagoras  from  A  KMN 
in  the  plan  (second  figure): 


KM  =  A/322  +  242  =  8  x  5  =  40. 
TanFMK  =  $g  =  1;  /.  ^FMK  =  45°. 


PLANE  TRIGONOMETRY 


357 


In  practical  problems,  boys  are  constantly  blundering 
over  compass  bearings.  Impress  on  the  class  that  the  difference 
between  the  bearings  of  two  distant  objects  is  the  angle  made 
by  the  two  lines,  drawn  in  the  Jf/.P.,  from  the  observer  to 
the  objects.  If  the  objects  are  above  the  H.P.,  the  difference 
between  the  bearings  is  still  an  angle  on  the  H.P.,  viz.  the 
angle  between  the  two  vertical  planes  drawn  through  the 
observer  and  each  of  the  objects. — An  observer  is  at  S  in  a 
H.P.,  his  south-north  line  being  SN.  PQ  and  RT  are  two 
vertical  poles.  He  takes  the  bearings  of  the  two  poles  and 


Fig.  199 

finds  that  their  horizontal  angles  are  respectively  N.  50°  W. 
and  N.  55°  E.,  i.e.  the  difference  between  their  bearings  is 
105°. 

Now  suppose  he  could  not  see  the  bottom  of  the  poles, 
because  of  an  intervening  hill.  The  observer  would  have 
to  point  his  telescope  at  the  pole-tops  P  and  R,  and  he  could 
then,  if  he  wished,  take  the  angles  of  elevation.  But  his  pur- 
pose now  is  to  take  the  difference  between  the  bearings,  and 
he  would  therefore  observe  where  each  vertical  plane  con- 
taining the  tilted  telescope  cut  the  horizontal  plane.  The 
angle  to  be  measured  (  ZQST)  would  be  exactly  the  same 
as  before.  Impress  on  the  boys  that  the  observer  could  not 
measure  the  angle  PSR  in  the  oblique  plane;  his  theodolite 
does  not  permit  of  that.  And  even  if  he  could,  the  angle 
would  not  represent  the  angle  between  the  bearings. 

Here  is  an  illustrative  problem.  Find  the  distance  between 
the  tops  of  the  spires  of  two  distant  inaccessible  churches.  (It 
would  be  taken  rather  later  in  the  course.) 


358 


CRAFTSMANSHIP  IN  MATHEMATICS 


Measure  off  a  base  line  AB  in  a  suitable  position,  and  from 
each  end  take  the  bearings  of  both  spires,  P  and  Q,  draw 


ELEVATION 
(V  P  *o.tl>  spines) 


a  ground  plan,  and  mark  in  the  Zs  a,  j8,  y,  8.    The 
f  may  be  calculated  if  wanted. 


and 


In  the  APAB,  AB  and  the  Zs  are  known; 

.*.  PA  and  PB  can  be  calculated. 
In  the  AQAB,  AB  and  the  Zs  are  known; 

.'.  BO  can  be  calculated. 
In  the  APBQ,  PB  and  BQ  are  calculated,  and  ZS  known, 

/.  PQ  can  be  calculated. 

Now  examine  the  perspective  sketch,  with  the  2  spires  P'P 
and  Q'Q  in  position.  We  have  to  find  P'Q'.  We  know  AP, 
BQ,  PQ.  Measure  the  /.s  of  elevation  p  and  a\  P'P  =  AP 
tan/),  Q'Q  =  BQ  tancr;  hence  P'P  and  Q'Q  are  known. 
Hence  in  the  elevation,  everything  is  known  except  P'Q', 
and  this  is  easily  calculated  by  Pythagoras. 

Make  the  boys  do  this  practically.    Any  two  distant  tall 
objects  will  do. 


PLANE  TRIGONOMETRY  369 


The  Obtuse  Angle 

Angles  up  to  180°  should  be  considered  at  an  early  stage, 
but,  before  angles  greater  than  180°  are  considered,  substantial 
progress  on  the  practical  side  of  the  subject  is  desirable. 

Remind  the  boys  that  the  rotating  arm  of  the  angle, 
regarded  as  the  hypotenuse  of  a  rt.  Zd  A,  may  be  carried 
round  from  90°  to  180°,  the  pivot  being  the  point  of  inter- 
section of  the  co-ordinate  axes.  Refer  to  the  work  on  graphs, 
and  the  rule  of  signs  for  the  second  quadrant;  all  x  values 
measured  to  the  left  of  the  origin  are  regarded  as  negative. 
If,  for  instance,  we  consider 
the  triangle  BOC  in  the 
second  quadrant,  the  hypo- 
tenuse and  perpendicular  are 
positive  as  in  the  first  quad- 
rant, but  the  base  OC  is 
negative.  Proof?  There  is 
none.  It  is  merely  an  accepted  Fig.  201 

convention. 

Suppose,  then,  we  have  an  angle  greater  than  90°,  say 
145°.  How  am  I  to  find  the  value  of  its  sine,  cosine,  and 
tangent? 

Exactly  as  before.  From  any  point  on  the  rotating 
arm  OB,  drop  a  perpendicular  to  the  fixed  arm  OA  (pro- 
duced backwards,  because  necessary),  and  take  the  ratios 
in  the  same  way  as  we  did  for  acute  angles.  But  remember 
the  signs.  For  this  angle  145°,  the  hypotenuse  is  OB,  the 
base  OC,  the  perpendicular  BC. 

TT  •        IJfO  BC  1JKO  OC  1    AtaO  BC 

Hence:  sin  145  = — ;  cos  145  = ,  tan  145  = 


OB'  ~"w  ™OB~ 

It  may  not  look  as  if  the  perpendicular  BC  concerned  the 
angle  AOB  (145°),  but  how  else  could  a  perpendicular  for 
145°  be  drawn? 

Now  consider  an  acute  angle  equal  to  the  angle  BOC 
in  fig.  201.    Evidently  it  is  (180°  -  145°)  or  35°.    Fig.  202 


360 


CRAFTSMANSHIP  IN  MATHEMATICS 


shows  OB'  equal  to  OB  in  fig.  201.     Hence  the  triangle 
B'OC'  has  sides  of  exactly  the  same  length  as  the  triangle 

TVP'  OP'  ' 

BOC.    SinS5'  =  fL;  cos35°  =         ;  tan35°  = 


Fig.  202 

Comparing  the  ratios  for  145°  and  35°,  we  see  that: 

sin35°  =  sinl45°, 
cos35°  =  _cos!45°, 
tan  35°-  -tan  145°. 

The  same  thing  must  apply  to  any  pair  of  angles  whose  sum 
is  180°.   Thus  we  may  say, 

sinA  =  sin(180°  -  A), 
cosA  =  -cos(180°  —  A), 
tanA  =  -tan(180°  -  A). 

If  the  above  demonstration  is  attempted  from  a  single 
figure  (as  it  might  well  be  from  fig.  201),  slower  boys  will 
inevitably  be  confused. 

It  is  an  excellent  plan  to  make  boys  in  Upper  Sets  express 
their  ratios  in  terms  of  co-ordinates,  i.e.  to  call  the  rotating 
arm  r,  and  its  extremity  P  (#,  y). 

Give  plenty  of  oral  practice  in  the  obtuse  angle  relations, 
e.g.  tanlOO0  =  — tan(180°  -  100°)  =  -tan80°  =  -5-67. 


PLANE  TRIGONOMETRY  361 


The  General  Triangle  and  its  Subsidiary 
Problems 

Before  proceeding  to  the  solution  of  triangles,  revise 
carefully  the  geometry  of  congruent  triangles,  and  note 
what  various  sets  of  data  are  necessary  and  sufficient  for 
copying  a  triangle.  A  triangle  is  determined  uniquely  if  we 
are  given  (1)  the  3  sides,  (2)  2  sides  and  the  included  angle, 
(3)  1  side  and  2  angles.  If  we  are  given  2  sides  and  the  angle 
opposite  one  of  them,  there  may  be  2  solutions,  or  1  solution, 
or  no  solution. 

A  triangle  cannot  be  determined  unless  the  data  include 
at  least  one  side. 

Thus  the  necessary  data  include  3  elements,  at  least 
one  of  them  being  a  length. 

All  the  formulae  in  this  section  must  be  established 
geometrically.  As  geometrical  exercises  they  are  all  first- 
rate. 

1.  In    any    triangle    ABC,   a  =  b  cosC  +  c  cosB. — Show 
that  this  relation  holds  good  for  both  acute-angled  and  obtuse- 
angled  triangles.    It  is  simply  a  question  of  dropping  a  per- 
pendicular   and    considering    separately    the    two    resulting 
right-angled  triangles. 

Do  not  forget  the  sister  expressions  in  this  and  sub- 
sequent formulae.  The  one  thing  to  keep  in  mind  is  the  cyclic 
order  of  the  letters  A,  B,  C,  and  a,  A,  c.  For  instance,  the 
above  identity  may  be  written, 

b  =  c  cos  A  +  a  cosC, 
or,    c  =  a  cosB  +  b  cos  A. 

2.  The  sine  formula. — In  any  triangle  ABC, 


sin  A      sinB 
As  before,  show  that  the  relation  holds  good  for 


sinC 
obtuse-angled  as  well  as  for  acute-angled  triangles. 


362  CRAFTSMANSHIP  IN  MATHEMATICS 

The  "  ambiguous  case  "  should  receive  special  attention. 
Link  up  the  work  with  the  closely  analogous  case  in  geometry. 
In  fact  the  problems  are  the  same.  Readily  understood  as 
they  generally  are,  they  are  often  half  forgotten.  They  must 
be  regarded  as  sufficiently  tedious  and  troublesome  as  to 
merit  special  and  repeated  attention. 

3.  The  cosine  formula. — In  any  triangle  ABC,  c*  =  a2  -f  ^2 
~-2ab  cosC.  Again  be  careful  to  consider  both  acute-angled 
and  obtuse-angled  triangles.  Link  up  carefully  with  Py- 
thagoras and  its  extensions  (Euclid,  I,  47;  II,  12,  13).  The 
solution  is  straightforward  and  seldom  gives  trouble. 

When  solving  triangles,  use  sine  or  cosine  formula? 


If  given  (1)  3  sides, 

or  (2)  2  sides  and  in-, 
eluded  angle, 


use  cosine  formula  for  first 
operation;  then  continue 
with  the  quicker  sine  for- 
mula, using  it  to  find  the 
smaller  of  the  two  remain- 
ing angles. 

If  given  (3)  2  angles  and  1  side,  }  use  sine 

or  (4)  2  sides  and  a  not-included  angle,)     formula. 

N.B.  (1)  If  given  3  sides,  find  the  smallest  angle  first. 

(2)  If  the  given  triangle  is  isosceles,  use  neither 
formula,  but  drop  a  perpendicular  to  the 

base. 

* 

4.   The  tangent  formula. — In  any  triangle  ABC  (where 
b>c), 

tanj-(B  —  C)  _  b  —  c 
tan|(B  +  C)  ~~  b  +  c 

This  is  a  useful  alternative,  more  suitable  for  log  calculations, 
when  2  sides  and  the  included  angle  are  given.  The  cos 
formula  is  often  cumbrous  in  application,  not  being  suitable 
for  log  calculation. 


PLANE  TRIGONOMETRY  363 

The  boys  must  learn  to  establish  the  formula  geometrically, 
from  first  principles,  and  not  derive  it  from  other  trigono- 
metrical formulae.  But  for  beginners  it  is  generally  puzzling. 
Begin  by  giving  them  a  particular  case  to  which  they  may 
apply  the  formula.  Let  the  sides  of  a  triangle  be,  say,  11, 
13,  16,  and  let  the  boys  work  out  the  angles  from  their  cos 


B  a   «  16  O 

Fig.  203 

and  sine  rules,  using  four-figure  tables.     The  angles  shown 
in  the  figure  are,  to  the  nearest  minute, 

tan|(B  -  C)  =  tan |(53°  47' -43°  3') 
tanJ(B  +  C)       tan|(53°  47'  +  43°  3') 

tan5°22;         -094         1 


Again: 


tan  48°  25'       1-127       12* 
b-c       13-11        1 


b  +  c       13  +  11       12 


Thus  they  see  that,  at  least  in  this  particular  case,  the  theorem 
holds  good.  Working  out  a  particular  case  in  this  way,  they 
grasp  the  fact  that  tan|(B  — C)  is,  after  all,  just  the  tan  of 
a  simple  angle.  So  with  |(B  +  C). 

The  problem  now  is  to  devise  a  figure  which  shall 
actually  show  these  angles  £(B  —  C)  and  f  (B  +  C);  also 
the  sum  (b  +  c)  and  the  difference  (b  —  c)  of  the  sides. 

There  are  two  subsidiary  points  to  note  first. 


364  CRAFTSMANSHIP  IN  MATHEMATICS 

(1)  In  any  triangle, 

since        A  +  B  +  C  =  180°, 


or 

Again: 

or 
Similarly: 


+ 


_      0 

-90     --. 


T' 


B  f  C 


B-C 


22 

t_A  _  A  =  C— 


Give  plenty  of  oral  work  on  these  points,  with  blackboard 
figures  to  illustrate. 

(2)   How  have  we  been  able  in  geometry  to  show  the 
sum  and  difference  of  two  sides  of  a  triangle? 


Fig.  204 


The  sum  of  b  and  c  may  be  shown  by  swinging  round  AB 
on  A  to  AE,  so  that  AE  =  AB;  hence  CE  =  (b  +  r);  the 
difference  may  be  shown  by  cutting  from  AC  a  part  AD  equal 


PLANE  TRIGONOMETRY 


365 


to  AB;  thus  DC  =  AC  —  AB  =  (b  —  c).  The  same  figure 
shows  £(B  —  C).  For  (fig.  ii),  since  m  —  n  +  jf>,  .".  m'  =  n  +  />> 
/.  m'  +  p  =  n  +  2p,  or  B  =-  C  +  2p;  ;.  £  ==  J(B  —  C). 

(i)  Now  we  may  draw  the  required  figure. 

With  centre  A  and  radius  AB,  describe  the  circle  EBD, 

and  produce  CA  to  E.    Evidently  EC  =  (b  +  c),  DC  =  (b  -  c). 

Join  EB.      Z.E  (at  circf.)  =  ^A  (at  centre).     We  know  that 

/.DEC  —  |(B  —  C),  but  there  seems  to  be  no  obviously 


Fig.  205 


simple  way  of  using  it.    But  if  we  draw  CF  parallel  to  BD, 
to  meet  EB  produced  in  F,    Z.BCF  =   ZDBC  =  £(B  —  C). 

Again,  in  the  right-angled  triangle  EFC,  since   /.E  =  £A, 
ZECF  =  1(B  +  C). 

Thus  we  have  the  two  angles  and  the  two  lengths  for  the 
tan  formula: 

BF 

tan^(B-C)  =  tanBCF  =  FC  =  BF  =  DC  _  b  —  c     Q  E  D 
tanJ(B  +  C)~~tanECF      EF      EF      EC       b  +  c  *    * 

FC 

(ii)  A  boy  might  very  well  ask  if  we  could  use  the  figures 


366  CRAFTSMANSHIP  IN  MATHEMATICS 

made  by  drawing  the  circle  with  radius  AC  instead  of  AB. 
Exterior    ZA  at  centre  =  B  +  C;     /.     ZD  at  circum- 
ference =  |(B  +  C). 


Fig.  206 

Also      /.BCD  =  ZB  —  LD     (ext.     L    property)  =  ZB 
-  |(B  +  C)  -  KB  -  C). 

We  may  take  the  tan  of  the  last  angle  by  dropping  the 
±  BF. 

BF 

tan£(B~C)  _  tanBCF  „  FC  _,  FD  _  BD  _  b-c    QED 
tan|(B  +  C)      tanBDF      BF      FC       BE       b  +  c  '    " 

FD 

There  is  no  essential  difference  between  the  two  proofs. 

(iii)  Or  a  boy  might  ask  if  we  could  not  derive  the  angle 
+  C)  from  the  JA  obtained  by  actually  bisecting  A. 


PLANE  TRIGONOMETRY 


367 


Let  AD  be  the  bisector,  and  let  CD  meet  it  at  right  angles. 
Draw  BF  perpendicular  to  CD  produced,  and  BE  perpen- 
dicular to  AD. 

Evidently    ZBCF 
=  KB  —  C),       and 
Z.AEE  =  ZACD  = 
KB  +  C). 

The  figure  does 
not  give  us  a  length 
AC  +  AB  (=  b  +  c), 
or  a  length  AC  —  AB 
(=  b  —  c).  But  we 

can  project  AB   and  Fig  20? 

AC  on  to   FC;    FD 

(=  BE)  is  the  projection  of  AB,  and  DC  is  the  projection 
of  AC,  and  so  we  may  obtain  what  we  want  in  this  way: 

(1)  AD  =  b  sinKB  +  C);  AE  ==  c  sin|(B  +  C); 
.-.  BF  =  AD  -  AE 


Vi(B-c) 


(2)  DC  =  b  cosKB  +  C);  FD  =  c  cos£(B  +  C): 

/.  FC  =  DC  +  FD 

=  (b  +  c)  cos|(B  +  C). 

.  BF  _  (b  -  c)  sinj(B  +  C)  _  b  -  c 
(6)  •  •  FC  ~  (4  +  c)  cos  KB  +  C)      V+c  a  i( 


But 


BF 
£  =  tani(B 


C); 


. 
b 


tan |(B  +  C)  =  tan  KB  -  C), 

b  —  c  =  tan  J(B  —  C) 
b  +  c  ~~  tan  KB  +  Cj" 


or 


This  last  method  is  not  quite  so  simple  as  the  first,  but  it 
appeals  to  A  Sets  to  whom  alone  (perhaps)  it  should  be  given, 


368  CRAFTSMANSHIP  IN  MATHEMATICS 

5.  Other  formulae  that  should  be  worked  out  geometrically: 

(i)  Cos2a  =  2cos2a  —  1  =  1  —  2  sin2a. 
(ii)  Sin2a  =  2  sin  a  cos  a. 

02     I     £2  _  C2 

(iii)  CosC  —  --  --  -  ----    (and   thence,    algebraically, 


.  -- 

the  half-angle  formula,  sin|A  ==  y  £  ~  *)  (*  ~  c) 

o     \  be          * 

&c.). 

(iv)  Area  =  \bc  sin  A;  &c. 
(v)  Circles    of    a    triangle:     circumscribed,    inscribed, 

escribed. 
(vi)  Medians,    angle    bisectors,   pedal    triangle,    ortho- 

centre,  &c. 

The  geometry  of  these  basic  formulae  is  the  important  thing. 
The  derivatives  may  be  obtained  algebraically. 

Angles  up  to  360°.    The  Four  Quadrants 

It  is  best  to  begin  by  showing  the  boys  how  surveyors 
in  their  work  often  find  it  an  advantage  to  consider  angles 
up  to  360°.  We  have  therefore  to  decide  how  the  ratios  of 
angles  between  180°  and  360°  can  be  expressed.  Thus  we 
have  to  consider  the  3rd  and  4th  quadrants. 

Remind  the  boys  that  there  are  no  proofs  of  our  conven- 
tions concerning  the  signs  in  the  four  quadrants.  The  con- 
ventions are  just  a  matter  of  convenience,  arrived  at  by 
general  consent,  and  consistent  with  one  another.  It  is  this 
consistency  which  is  the  important  thing.  The  boys  must 
be  drilled  in  the  quadrant  signs  until  the  last  shred  of  doubt 
disappears. 


44  Plus:  right  and  above," 
44  Minus:  left  and  below." 

Fig.  308 


PLANE  TRIGONOMETRY 


369 


Other  important  memos. 

1.  The  fixed  arm  of  the  angle  is  always  in  the  3  o'clock 
position. 

2.  The  rotating  arm  of  the  angle  always  moves  counter- 
clockwise. 

3.  Never  take  a  short  cut  by  moving  clockwise. 

From  any  point  in  the  rotating  arm  we  may  drop  a  per- 
pendi  cular  PM  on  the  abscissa,  form  a  right-angled  triangle, 


M. 


(0 


(ii) 


Fig.  209 


and  so  take  any  ratio  of  any  angle  in  any  quadrant.    Taking 
e.g.,  the  tangent,  we  have: 


— OM2 
tanXOP,  =  d^»  = 


tanXOP  = 
ta  4 


~ 
+OM4 


OM2' 


OM' 


-. 
OM4 


(1291) 


25 


370 


CRAFTSMANSHIP  IN  MATHEMATICS 


Beginners  are  often  puzzled  about  the  re-entrant  angles  in 
the  3rd  and  4th  quadrants.  Make  them  understand  that  if 
they  take  the  smaller  angles  in  these  quadrants,  they  have 
taken  a  clockwise  rotation  of  the  moving  arm,  and  this  is 
not  allowable.  (Postpone  the  consideration  of  negative  rota- 
tions until  the  main  principle  is  grasped  thoroughly.) 

As  already  suggested,  an  alternative  plan  is  to  call  the 
length  of  the  rotating  arm  r,  and  to  call  the  point  P  which 
we  fix  in  it  (x,  y),  x  and  y  being  the  co-ordinates  of  the  point. 
But  if  the  boys  are  at  first  well  drilled  in  the  use  of  the  terms 
hypotenuse,  base,  and  perpendicular,  these  terms  will  probably 
continue  to  be  used,  at  least  mentally.  In  A  Sets,  the  co- 
ordinate notation  is  preferable:  its  advantages  are  obvious. 

Make  the  boys  memorize  the  following  scheme:  it  merely 
amplifies  what  was  said  on  a  previous  page. 


i. 

2. 

3. 

4. 

Sin      .  . 

+ 

+ 





Cos     .. 

-f- 

— 

— 

+ 

Tan     .  . 

+ 

— 

+ 

— 

"  Sin,  cos,  and  tan  are  +  in  the  1st  quadrant,  and  each  is  +  in 
one  other,  viz.  sin  in  2nd,  tan  in  3rd,  cos  in  4th."* 

Give  plenty  of  oral  work  on  the  ratios  of  angles  in  all 
four  quadrants.  Boys  should  recognize  the  landmarks  90°, 
180°,  270°,  360°,  and  know  at  once  in  which  quadrant  a 
given  angle  occurs. 

Beginners  are  often  caught:  they  take  the  complement 
of  the  angle  instead  of  the  angle  itself.  Point  out  again  and 
again  that  whatever  angle  we  may  have  in  the  first  quadrant 
there  must  be  angles  with  exactly  the  same  numerical  ratio 
in  the  other  three  quadrants.  The  four  resulting  triangles 


*  One  or  two  schools  use  this  mnemonic: 
s  *  |- all 
T      I 


"  positively  all  silver  tea  cups  ". 


PLANE  TRIGONOMETRY 


371 


formed  by  dropping  a  perpendicular  from  the  same  point 
P  on  the  rotating  arm  must  be  congruent. 


/               \ 

N^ 

1 

+  1-28                +1-28 

> 

0 

"*" 

0                    X 

as: 

232 


(iii) 


Fig.  210 


Note  the  4  angles:  52°;  180°  -  52°  =  128°;  180°  +  52° 
=  232°;   360°  -  52°  =  308°. 

1*28 
Note  also  the  4  tangents:*   tan52°  =  +  -  ;    tan!28° 


=  -         ;  tan232°  = 


;  tan308°  =  - 


The  angle  in  the  second  quadrant  is  obtained  by  sub- 
tracting 52°  from  180°.  - 

The  angle  in  the  third  quadrant  is  obtained  by  adding 
52°  to  180°. 

The  angle  in  the  fourth  quadrant  is  obtained  by  sub- 
tracting 52°  from  360°. 


•Quite  by  chance  the  angle  in  the  second  quadrant  (128°)  has  the  appearance  of 
being  100  times  the  value  of  the  tangent  (1-28). 


372 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  four  angles  do  not  form  an  arithmetical  progression, 
and  they  cannot  do  so  unless  the  angle  in  the  first  quadrant 
is  45°. 

Any  such  group  of  4  angles  forms  a  symmetrical  figure: 


Whenever  we  take  a  trigonometrical  ratio  of  an  angle 
from  the  tables,  the  angle  is  one  belonging  to  the  first  quad- 
rant. But  there  are  three  other  angles  having  the  same 
numerical  value.  If  a  is  the  angle  in  the  first  quadrant,  the 
other  three  are  180°  -  a,  180°  +  a,  360°  —  a.  But  each 
ratio  in  each  quadrant  has  its  own  signs  as  we  have  already 
seen. 

To  evaluate  the  ratios  of  angles  greater  than  90°,  we  may 
remember  the  formula,  «180°  ±  a,  though  this  is  really 
intended  to  include  angles  greater  than  360°.  Let  the  boys 
make  up  this  general  formula  from  an  examination  of  a 
number  of  particular  cases. 


First  Notions  of  Periodicity 

The  boys  are  already  familiar  with  the  notion  that  the 
rotating  arm  of  the  angle  may  proceed  beyond  one  revolution; 
the  movement  of  the  pedal  of  an  ordinary  bicycle  serves  to 
convey  the  notion  of  angles  of  w360°  or  w360°  +  a.  Show 
clearly  that  the  ratios  of  any  angle  a  are  exactly  the  same  as 
those  of  any  angle  that  differs  from  a  by  any  number  of 


PLANE  TRIGONOMETRY 


373 


complete  revolutions.  Thus,  sin(w360°  +  a)  —  sin  a,  where 
n  is  any  integer;  so  with  all  the  ratios.  For  example, 

cos 700°  =  cos (720°  -  20°)  -  cos (2  .  360°  —  20°) 
=  cos  —  20°  =  cos  20°. 

Give  ample  oral  practice  to  emphasize  the  fact  that  the  addition 
or  subtraction  of  any  multiple  of  360°  does  not  alter  the  value 
of  any  ratio  of  an  angle.  The  general  rule  may  be  expressed: 
"  If  a  is  an  angle,  any  ratio  of  %nir  i  a  is  numerically  equal 
to  the  same  ratio  of  a  ".  The  sign  to  be  attached  depends 
on  the  quadrant.  (The  radian  notation  should  be  familiar 
by  this  time.) 

For  purposes  of  illustrating  continuous  functions,  graphs 
may  be  obtained,  with  sufficient  accuracy,  by  30°  and  60° 


360     0 


360     0 


Fig.  212 

parallels  and  perpendiculars  as  in  fig.  183.  The  two  inter- 
mediate points  in  each  quadrant  are  enough  to  determine 
the  curve  fairly  readily.  The  boys  should  be  able  to  sketch 
the  curves  rapidly  and  should  become  thoroughly  familiar 
with  them.  They  should  note  that  if  the  graph  of  cos  a  be 
moved  90  units  along  the  x  axis,  it  coincides  with  that  of 
sin  a,  and  that  this  is  equivalent  to  saying  that  sin  (a  +  90°) 
=  cos  a.*  Superpose  the  cos  graph  on  the  sine  graph  and 
discuss  the  intersecting  points  and  the  ratios  of  the  angles 
indicated  by  those  points.  Draw  a  continuous  sine  graph  up 
to  5?r  or  GTT.  Select  some  first  quadrant  angle,  say  40°, 
raise  a  perpendicular  to  cut  the  graph,  and  through  the 
point  of  intersection  run  a  parallel  to  the  x  axis  and  another 
the  same  distance  below  the  axis.  Discuss  and  compare  the 


•Slower  boys  will  confuse  90°  -f  A  with   180°  —  A. 
necessarily  different  unless  A  =  45°. 


Show  that  they  are 


374 


CRAFTSMANSHIP  IN  MATHEMATICS 


sines  of  all  the  angles  indicated  by  the  successive  points 
of  intersection.  Show  clearly  that  there  is  a  period  of  27r, 
and  that  sin#  may  therefore  suitably  be  called  a  periodic 
function  of  x.  So  with  cos  x.  Tan  x  is  likewise  a  periodic 


7 


~3ir 


\ 


lig.  213 


function,  but  with  a  period  of  TT  (not  2??);   show  how  this 
may  be  inferred  from  the  parallel  tan  curves. 


Compound  Angles 

1.  Sin(A  +  B)  =  sinA  cosB  +  cosA  sinB. 

2.  Cos(A  +  B)  =  cosA  cosB  —  sinA  sinB. 

3.  Sin(A  —  B)  =  sinA  cosB  —  cos  A  sinB. 

4.  Cos(A  —  B)  —  cos  A  cosB  +  sinA  sinB. 

Beginners  naturally  think  that  sin  50°  =  sin  20°  +  sin  30°, 
that  cos 70°  =  cos80°  —  cos  10°.  Give  a  few  examples,  with 
free  reference  to  the  four-figure  tables,  to  show  that  this  is 
not  so. 

Of  the  four  identities  named  above,  at  least  one  should 
be  proved  geometrically  and  mastered  thoroughly.  The 
neatest  method  is  the  projection  method,  and  with  A  Sets 
the  general  case  can  readily  be  proved  by  this  method.  With 
B  Sets  and  certainly  with  C  Sets  the  problem  is  best  con- 
sidered merely  from  the  point  of  view  of  positive  acute 
angles.  All  the  books  give  the  solution,  but  the  boys  should 
be  taught  to  analyse  the  conditions  of  the  problem,  not 
merely  to  follow  out  a  book  solution. 

The  following  sequence  of  arguments  is  suitable  for  teach- 
ing purposes. 


PLANE  TRIGONOMETRY 


375 


Let  OX  rotate  through  ZA  to  OC,  then  through  ZB 
to  OD;  in  its  complete  journey  to  OD  it  has  rotated  through 
the  complete  L  (A  +  B).  We  have  to  prove  that  sin  (A  +  B) 
=  sin  A  cos  B  +  cos  A  sin  B,  and  in  connexion  with  the 
three  angles  this  means  the  consideration  of  five  ratios,  viz. 


Fig.  214 


the  sines  of  A,  B,  and  A  +  B,  and  the  cosines  of  A  and  B. 
We  will  try  to  show  all  these  in  one  figure. 

Evidently  we  require  three  perpendiculars,  since  there 
are  three  angles. 

(1)  From  any  point  P  in  OD,  drop  the    JL  PN  to  OX. 
The  sine  of  L(A  +  B)  may  be  considered  from  the  rt.  Zd  A 
PON. 

(2)  From  P,  drop  a  -L  PQ  on  OC.   The  sine  and  the  cos  of 
ZB  may  be  considered  from  the  rt.  Zd  A  POQ. 

(3)  From  Q,  drop  a  -L  QR  on  OX.   The  sine  and  cos  of 
ZA  may  be  considered  from  the  rt.  Zd  A  QOR. 

When  we  have  to  prove  that  a  simple  expression  is  equal 
to  a  more  complex  expression,  it  is  a  good  general  rule  to 
begin  with  the  latter,  try  to  simplify  it,  and  get  back  to  the 
former.  Thus  we  may  begin: 

sin  A  cosB  -f-  cos  A  sinB 

=  cm  QO    OR  PQ 

QO  '  OP  +  QO  '  OP' 
But  how  are  we  to  proceed  now?    True  the  OQ's  seem  to 


376 


CRAFTSMANSHIP  IN  MATHEMATICS 


cancel  out  in  the  left-hand  term,  but  we  do  not  seem  to  be 
able  to  simplify  any  further. 

Since  a  circle  will  go  round  ONQP  (on  OP  as  diameter), 
Z.MPQ  =   ZA.     If  then  we  draw  QM    ±   PN,  we  have  a 


N    R  X 


Fig.  215 


APMQ  similar  to  AQRO.  Thus,  as  far  as  ratios  are  con- 
cerned we  may  consider  APMQ  instead  of  AQRO.  Now 
let  us  try  simplification  again: 

sin  A  cosB  -f  cos  A  sinB 


Q0  ,   PM  PQ 
'OP       PQ'OP 


QR    .  PM      PN        .   /A   ,  m 

c=  -  —  +  -  =  -  =  sin  (A  +  B). 
OP      OP       OP  V          J 

We  may  now  set  our  proof  as  an  examiner  would  expect  to 
see  it. 

QR      PM 
— 


_  QR   OQ      PM    PQ 
"~OP'OQ       OP  'PQ 

(each  term  multiplied  by  1) 

=  QR    OQ      PM    PQ 
OQ  '  OP  +  PQ  '  OP 

=  sinA  sinB  +  cosA  sinB.      Q.E.D. 


PLANE  TRIGONOMETRY 


377 


The  three  analogues  now  follow  on  simply.  All  four  identities 
should  be  verified  by  a  few  particular  cases  (4-figure  logs  will 
do),  e.g. 

(sin55°  cos 25°  +  cos 55°  sin 25°, 


sin  80°  = 


IsinlO0  cos  70°  +  cos  70°  sin  10°. 


Fig.  216 


Some  teachers  prefer  this  proof  instead: 

Let  the  acute  Z.s  A  and  B 
be  the  Zs  of  a  A  ABC.  Draw 
a  circle  round  the  A,  and  the 
radii  OA,  OB,  OC.  Evidently 
ZAOC  -  2B,  ZBOC  =  2A. 
If  J-s  from  the  centre  be 
drawn,  they  bisect  the  sides 
of  the  A .  Hence,  AB  =  d  sin 
(A  +  B),  AC -rf  sinB,  CB 
=  rfsinA;  also  AB  =  AC 
cos  A  +  CB  cosB.  By  equat- 
ing the  first  and  last  of 
these,  and  substituting  from  the  2nd  and  3rd: 

d  sin(A  +  B)  =  AC  cos  A  +  CB  cosB 

=  d  sinB  cos  A  +  d  sin  A  cosB, 
or    sin(A  +  B)  —  sinA  cosB  +  cosA  sinB. 

This  proof  does  not  seem  to  appeal  to  boys  so  readily  as  the 
former  does. 

We  now  come  to  the  general  case.  B  and  C  Sets  find  it 
difficult,  and  as  a  rule  it  should  be  given  to  A  Sets  only. 
Of  the  various  methods  of  proof  the  two  following  are  the 
simplest  for  teaching  purposes. 

1.  The  Projection  Method. — This  method  is  productive 
of  mistakes  unless  the  boys  have  mastered  the  elementary 
principles  of  projection. 

Give  the  class  one  or  two  preliminary  exercises  of  the 
following  kind: 


378 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  angle  A  of  the  regular  pentagon  ABODE  touches  the 

X  axis,  with  which  AB  makes 
an  angle  of  12°.  Find  (1)  the 
horizontal  distance  of  the  ver- 
tex D  from  A,  and  (2)  the 
height  of  D  above  the  X  axis. 
(1)  Horizontal  distance  of 
D  from  A  =  projection  of 
AB  +  projection  of  BC  + 
projection  of  CD.  Remem- 
ber that  projection  means 
projection  with  proper  sign 
attached,  and  we  must  take 
the  Zs  which  AB,  BC,  CD 
make  with  the  +  direction  of  OX.  Take  AB  as  unity. 

/.  Distance  AH  -  AB  cos  12°  +  BC  cos  84°  +  CD  cos  156° 
=  -9781"  +  -1045"  —  -9135" 
=  •1691". 

(2)  Height  of  D  above  OX. 

Height  HD  -  OM  +  MN  +  NP 

=  AB  sin!2°  +  BC  sin84°  +  CD  sin!56° 
=  -2079"  +  -994:5"  +  4067" 
=  1-6091". 

Now  we  come  to  the  identity  sin(A  +  B)  =  sin  A  cosB  +  cos  A 
sinB.     Let  the   Zs  be  the  same  as  in  fig.  214.     From  any 


Fig.  218 


PLANE  TRIGONOMETRY 


379 


point  P  in  OD,  draw  PQ   -L  OC.    Project  the  three  sides  of 
the  APOQ  on  the  Y  axis. 

OM  =  ON  +  NM, 

.'.  projection  of  OP  =  sum  of  projections  of  OQ  and  QP. 

/.  OP  sinXOP  -  OQ  sinXOC  +  QP  cosXOC. 
.'.  OP  sin(A  +  B)  —  OP  cosB  sinA  +  OP  sinB  cosA, 

(OQ  =  OP  cosB,  QP  =  OP  sinB) 

i.e.  sin  (A  +  B)  =  sin  A  cosB  +  sinB  cos  A. 

In  a  similar  manner,  by  projecting  the  three  sides  on  the 
X  axis,  we  may  prove  that 

cos  (A  +  B)  =  cos  A  cosB  —  sin  A  sinB. 

Note  that  the  method  is  perfectly  general,  being  applicable 
to  any  angles. 

2.   The   Cosine   Rule   Method.— This   is   based   on   the 

rules  (1)  that  cos  A  =  — +  c  ~~  a_  and  ^2)  that  if  P  and  Q 

2bc 

are  the  two  co-ordinate  points   (xl9  jyt),  (x2,  j>2),  then  PQ2 

The  identity  usually  considered  is  cos  (A  —  B)  =  cos  A 
cosB  +  sin  A  sinB,  the 
others  being  treated  as 
derivatives. 

Whatever  two  angles 
are  given,  the  initial  line 
for  each  is  the  positive 
direction  of  the  X  axis. 
Note  that  we  are  taking 
the  difference  between 
two  angles,  not  their 
sum.  No  matter  what  (CosA,s^A) 
two  angles  are  taken,  Fig.  219 

cosPOQ  =  cos(A  — B). 

The  simplest  way  is  to  take  a  circle  of  unit  radius,  and 


Q(COS 


380  CRAFTSMANSHIP  IN  MATHEMATICS 

to  let  the  co-ordinates  of  P  be  (cos  A,  sin  A)  and  of  Q,  (cosB, 
sinB).  Note  that  OP  =  OQ  =  1;  hence  the  denominators 
of  the  cosine  ratios  need  not  be  written. 

Cos  (A-  B) 

=  cosPOQ 

=  Qp2  +  °Q2  -  PQ2 

2OP.OQ 
_  2  -  PQ2 

2 

2  -  ((cos  A  -  cosB)2  +  (sinA-sinB)2} 

. 


—  —  2cosAcosB  +  sin2A  +  sin2B—  2sinAsinB} 

2 

—  2  —  {2  —  (2  cos  A  cos  B  +  2  sin  A  sinB)  } 

2 
=  cos  A  cosB  +  sin  A  sinB. 

There  does  not  seem  to  be  much  to  choose  between  this 
method  and  the  project!  ve  method.  To  able  boys  both 
methods  appeal.  To  boys  of  poor  mathematical  ability, 
both  methods  are  equally  hateful. 

One  or  other  of  the  four  identities,  preferably  the  one 
proved  by  the  projective  method,  should  be  regarded  as 
basic,  and  the  others  should  be  treated  as  derivatives. 

Other  necessary  derivatives  are: 

(1)  2  sin  A  cos  B  =  sin  (A  +  B)  +  sin  (A  —  B),     and     its 
three  analogues. 

C  +  D       C  —  D 

(2)  sinC  +  sinD  =  2  sin—  —  —  cos  -  ,  and  its  three 

analogues. 

(3)  tan(A  +  B)  =   tanA  +  tanB 
v  '        v      i      y 


/A\  ±         OA 

(4)  tan  2  A  =  -  . 
v  '                  1  —  tan2A 

(5)  sin3A  =  3  sin  A  —  4  sin3A. 

(6)  cosSA  =  —3  cosA  +  4  cos8A. 


PLANE  TRIGONOMETRY  381 

All  the  formulae  should  be  learnt  off.    If  mnemonics  can  be 
devised,  they  will  help  the  lame  ducks  much. 

Let  the  boys  verify  all  the  formulae  established,  by  means 
of  a  few  simple  exercises.  Use  four-figure  logs  for  this  purpose, 
and  so  cover  a  good  deal  of  ground  in  a  short  time.  For 
instance,  show  that 

2  sin  50°  cos  24°  =  sin  (50°  +  24°)  +  sin  (50°  —  24°) 

=  sin74°  +  sin26°, 
(2  X  -7660  X  -9135)  =  (-9613  +  -4384),  &c. 

Books  to  consult: 

1.  Trigonometry,  Siddons  and  Hughes. 

2.  Elementary  Trigonometry,  Durell  and  Wright. 

3.  Advanced  Trigonometry,  Durell  and  Robson. 

4.  The  Teaching  of  Algebra,  Nunn. 

5.  Elementary  Trigonometry,  Heath. 

6.  Trigonometry,  Lachlan  and  Fletcher. 

7.  A  Treatise  on  Plane  Trigonometry,  Hobson. 


CHAPTER  XXVII 

Spherical    Trigonometry 

Spherical  trigonometry  enters  into  the  work  of  the  map- 
maker,  the  navigator,  and  the  astronomer;  also  into  the 
work  of  the  surveyor  if  that  work  extends  over  larger  areas, 
as  in  the  case  of  the  Ordnance  Survey.  But  for  an  under- 
standing of  the  essentials  of  surveying,  map-making,  navigation, 
and  astronomy,  little  more  than  the  A,  B,  C  of  spherical 
trigonometry  is  required,  and  all  this  can  be  included  in  a 
very  few  lessons.  The  elementary  geometry  of  the  sphere 
should  already  have  been  done. 

The  following  are  the  chief  points  for  inclusion  in  the 
necessary  elementary  course.  (Many  of  the  difficulties  can 


382  CRAFTSMANSHIP  IN  MATHEMATICS 

be  elucidated  by  the  use  of  simple  illustrations.  The  orange, 
with  its  natural  sections,  is  very  useful.  Well-shaped  apples 
lend  themselves  to  the  making  of  useful  sections.  A  slated 
sphere,  mounted,  should  always  be  available). 

1.  Great  and  small  circles. 

2.  Shortest    distance   that   can   be   traced   between   two 
points  on  the  surface  of  a  sphere — the  arc  of  the  great  circle 
passing  through  them.     (A  simple  experimental  verification 
is  good  enough  for  beginners.) 

A  suitable  argument:  If  a  string  be  stretched  between 
two  points  on  the  surface  of  a  sphere,  it  will  evidently  be 
the  shortest  distance  that  can  be  traced  on  the  surface  between 
the  points,  since,  by  pulling  the  ends  of  the  string,  its  length 
between  the  points  will  be  shortened  as  much  as  the  surface 
will  permit.  Any  part  of  the  stretched  string,  being  acted 
on  by  two  terminal  tensions,  and  by  the  reaction  of  the 
surface  which  is  everywhere  normal  to  it,  must  lie  in  a  plane 
containing  the  normal  to  the  surface.  Hence  the  plane  of 
the  string  contains  the  normals  to  the  surface  at  all  points 
of  its  length,  i.e.  the  string  lies  in  a  great  circle.  (Sixth 
Form  boys  ought  to  appreciate  such  an  argument.) 

3.  Axes;  pole  and  polar. 

4.  Primary  and  secondary  circles. 

5.  The  angle  between  two  great  circles  is  measured  by: 

(i)  the  angle  between  their  planes, 

(ii)  the  arc  intercepted  by  them  on  the  great  circle 

to  which  they  are  secondaries, 
(iii)  the  angular  distances  between  their  poles. 

6.  The  spherical  triangle — that  portion  of  the  surface  of  a 
sphere  bounded  by  the  arcs  of  three  great  circles.     Parts: 
3  sides  and  3  angles. 

7.  Since  3  great  circles  intersect  one  another  to  form  8 
triangles,  that  particular  triangle  is  selected  which  has  2,  or 
if  possible  3,  sides  each  less  than  a  quadrant. 

Cut  an  orange  or  an  apple  into  two  equal  parts; 


SPHERICAL  TRIGONOMETRY  383 

hold  the  two  parts  together,  and  cut  again  into  two 
equal  parts,  this  time  by  a  plane  oblique  to  the  first; 
hold  the  four  parts  together,  and  cut  still  again  into 
two  equal  parts,  by  a  plane  oblique  to  both  of  the 
other  planes. 

8.  The  analogy  between  theorems  in  plane  and  spherical 
trigonometry,  e.g.  any  two  sides  of  a  triangle  are  together 
greater  than  the  third. 

9.  Polar  triangles,  i.e.  triangles  so  related  that  the  vertices 
of  the  one  are  the  poles  of  the  sides  of  the  other. 

10.  Angular  limits  of  the  sides  and  angles  of  a  spherical 
triangle. 

11.  Fundamental  formulae: 

(i)  Any  spherical  triangle: 

.        cos  a  —  cosi  cose  ,      ,        7         x 

cos  A  = : — - — . (and  analogues). 

sm0  sine 

(ii)  Right-angled  triangles: 

sin  A  —  sin  a/sine; 
cosA  =  tani/tanc; 
tan  A  =  tanfl/sini. 

(iii)  Sine  rule: 

sin  A       sinB       sinC  2n 


sin  a        sin  b        sine       sin  a  sin  b  sine* 

All  the  proofs  are  simple.   The  only  trouble  is  in  the  drawing 
of  suitable  figures. 

12.  The  Latitude  problem.  This  is  perhaps  the  most 
important  of  the  elementary  problems  of  the  sphere. 

The  navigator's  "  dead  reckoning  "  depends  on  his  know- 
ledge of  two  things:  (1)  his  course  (direction),  (2)  the  distance 
run  (determined  by  log).  He  has  to  resolve  his  distance- 
course  into  separate  mileage  components  of  northing  and 
southing,  easting  and  westing.  Then  he  has  to  convert  his 


384  CRAFTSMANSHIP  IN   MATHEMATICS 

northing  and  southing  mileage  into  degrees  and  minutes  of 
latitude,  his  easting  and  westing  into  degrees  and  minutes  of 
longitude. 

There  is  no  difficulty  with  the  former.  The  meridians 
of  longitude  are  all  great  circles.  When  we  know  the  length 
of  the  circumference  of  these  circles,  a  simple  calculation 
will  give  the  change  of  latitude  produced  by  a  given  northing 
and  southing.  (Polar  circumference  —  24,856  miles;  therefore 
length  of  degree  of  latitude  =  69  miles;  -616  of  69  miles 
=  nautical  or  sea  mile  =  6080  ft.  Thus  60  sea  miles  —  1 
degree  of  latitude,  and  1  sea  mile  =  1  minute  of  latitude. 
"  Knots  "  =  sea  miles  per  hour.) 

But  parallels  of  latitude  are  small  circles  decreasing  from 
the  equator  to  the  poles.  Only  along  the  equator  itself  does 
1  sea  mile  imply  1  degree  of  longitude. — We  have  to  discover 
a  law  which  the  length  of  a  degree  of 
longitude  follows. 

This  law  does  not  show  a  length 
proportional  to  the  distance  from  the 
pole.  The  greatest  distance  between 
two  meridians  is  not  halved  at  45°, 
but  at  60°.  Why  has  the  parallel  of 
60°  half  the  circumference  of  the 
equator? 

Fig.  220  CE  =  radius;    A  =  point    in    lat. 

60°.     Let  figure  rotate  on  PP'.    The 

circle  will  trace  out  the  surface  of  the  globe,  E  will  trace 
out  the  equator,  and  A  the  parallel  of  60°  of  which  AB 
is  the  radius. 

CE  -  CA  =  R  (say). 
Then    AB  =  AC  sin  ACB, 
=  Rcos60°, 
=  *R. 

Since  AB  =  |R,  circf.  of  the  60°  parallel  =  \  length  of 
equator,  .*.  the  length  of  a  degree  in  60°  lat.  is  half  the  length 
of  a  degree  along  the  equator. 

Thus  a  voyage  of  a  given  number  of  sea-miles  along  the 


SPHERICAL  TRIGONOMETRY 


385 


60th  parallel  implies  a  change  of  longitude  twice  as  great 
as  the  same  distance  along  the  equator. — With  the  help  of 
the  slated  globe,  show  the  class  how  short  the  degrees  of 
longitude  necessarily  are  in  the  neighbourhood  of  the  Pole. 

We  give  a  suitable  figure*  for  showing  the  general  case. 
If  R  be  the  radius  of  the  equator,  and  r  the  radius  of  the 


parallel  of  latitude  A,  passing  through  a  given  point,  then 
r  =  R  cosA. 

Q  =  Lat.  0°,  long.  0°. 

P'  =  Lat.  X,  Long.  0°. 

V  =  Lat.  X,  Long.  P'V  west. 

T  =  Lat   0°,  Long.  QT  west. 
Difference  of  longitude  of  M  and  V  =  arc  MV  =  Z.MKV  =  LI. 

Give  the  boys  the  little  problem  to  prove  that  the  length 
of  1  minute  of  longitude  measured  along  a  parallel  of  lati- 
tude A  is,  1  nautical  mile  X  cosX. 

*  The  figure  is  designed  to  show  merely  the  main  geometrical  facts.  When  the 
boys  are  familiar  with  these  facts,  the  correct  notation  of  polar  co-ordinates,  and  the 
accepted  astronomical  sign  convention,  should  be  introduced. 

(E291)  26 


386 


CRAFTSMANSHIP  IN  MATHEMATICS 


It  requires  very  little  skill  in  soldering  to  make  a  wire 
model,  and  then  the  demonstration  is  exceedingly  simple. 

In  spherical  geometry  and  trigonometry,  good  figures  are 
essential,  or  very  few  boys  will  understand  the  problems 
considered.  Here  is  an  example  of  a  problem  from  one  of 
our  very  best  books  on  the  subject.  We  reproduce  the  original 

figure. 

The  excess  of  the  sum  of  the 
three  angles  of  a  spherical  triangle 
over  two  right  angles  is  a  measure 
of  its  area. 

Let  ABC  be  a  spherical  tri- 
angle; then,  since  the  sum  of  the 
three  spherical  segments  (lunes) 
ABA'C,  A'BC'B',  ACBC',  ex- 
ceeds the  hemisphere  ACA'  by 
the  two  triangles  ABC,  A'B'C'; 
and  since, 

(i)   the  measures  of  the  three 
spherical    segments    are,    respec- 
tively, the  angles  A,  B,  C,  of  the  spherical  triangle, 
(ii)  the  measure  of  the  hemisphere  is  2  right  angles, 

/.  the  sum  of  the  three  angles  exceeds  2  right  angles.    .    (i) 

If  A  is  the  number  of  degrees  in  the  angle  A,  S  the  surface 

A 

of  the  hemisphere,  the  area  of  the  spherical  segment  — .  S; 

180° 

.'.  since  ABC  is  equal  to  its  symmetric  triangle  A'B'C',  the 
result  of  (i)  is  that  if  S  is  the  area  of  the  spherical  triangle, 


/A  +  B  +  C 
I        180 


=  2S, 


or 


L  =  A  +  B  +  C  --  180 

360  '    ' 


i.e.  the  area  S  is  proportional  to  the  excess  of  A  +  B  +  C 
over  2  rt.  Zs. 


SPHERICAL  TRIGONOMETRY 


387 


I  have  given  this  theorem  to  boys  on  several  occasions, 
but  they  have  almost  invariably  failed  to  visualize  the  figure 
properly.  They  failed  to  pick  out  the  spherical  segments. 
We  append  four  new  figures.  The  first  shows  the  spherical 
triangle  plainly;  the  next  three  show  the  three  lunes,  separately 
shaded.  The  real  trouble  is  that  half  the  second  lune  (iii), 
viz.  the  part  A'B'C'  (=  the  symmetric  triangle  of  ABC)  is 
not  visible.  When  the  shaded  lunes  of  ii,  iii,  iv  are  added 


together,  it  is  seen  that  the  A  ABC  is  included  twice  and 
the  hidden  A'B'C'  once.  Hence  the  sum  of  the  three  shaded 
areas  exceeds  the  hemisphere  by  the  two  triangles  ABC 
and  A'B'C'. 

Books  to  consult: 

1.  Spherical  Trigonometry,  Murray. 

2.  Practical  Surveying  and  Elementary  Geodesy,  Adams. 


CHAPTER  XXVIII 

Towards  De   Moivre.      Imaginaries 


Interpretation  of  V  —  1 

"  Please  sir,  what  is  the  good  of  De  Moivre's  theorem? 
What  is  it  really  all  about?  What  is  the  use  of  talking  about 
imaginary  roots  to  equations?" 


388  CRAFTSMANSHIP  IN  MATHEMATICS 

Thoughtful  boys  often  ask  such  questions.  It  is  our  business 
to  see  that  our  answers  satisfy  them. 

The  symbol  V—  1,  if  interpreted  as  a  number,  has  no 
meaning.  But  algebraic  transformations  which  involve  the 
use  of  complex  quantities  of  the  form  a  +  bi  (where  a  and 
b  are  numbers,  and  /=  V— 1)  yield  propositions  which  do 
relate  purely  to  numbers,  and  those  propositions  are  now 
known  to  be  rational  and  acceptable 

Boys  should  understand  that  algebra  does  not  depend  on 
arithmetic  for  the  validity  of  its  laws  of  transformation.  If 
there  were  such  a  dependence,  it  is  obvious  that  as  soon 
as  algebraic  expressions  are  arithmetically  unintelligible,  all 
laws  respecting  them  lose  their  validity.  But  the  laws  of 
algebra,  though  suggested  by  arithmetic,  do  not  depend  on 
it.  The  laws  regulating  the  manipulation  of  algebraic  symbols 
are  identical  with  those  of  arithmetic,  and  it  therefore  follows 
that  no  algebraic  theorem  can  ever  contradict  any  result 
which  could  be  arrived  at  by  arithmetic,  for  the  reasoning 
in  both  cases  merely  applies  the  same  general  laws  to  different 
classes  of  things.  If  an  algebraic  theorem  is  interpretable 
in  arithmetic,  the  corresponding  arithmetical  theorem  is 
therefore  true.  Sixth  Form  boys  seem  to  gain  confidence 
when  once  they  realize  that  algebra  may  be  conceived  as 
an  independent  science  dealing  with  the  relations  of  certain 
marks  conditioned  by  the  observance  of  certain  conventional 
laws. 

It  is  true  that  the  present-day  use  of  imaginary  quantities, 
in  accordance  with  the  authoritative  interpretation  now 
given  them,  does  not  involve  any  sort  of  contradiction  and  is 
therefore  presumably  valid,  for  absence  of  logical  contradic- 
tion is  certainly  a  good  test  of  valid  reasoning.  But  Mr. 
Bertrand  Russell  is  perhaps  going  a  little  far  when  he 
says  (Prin.  of  Maths.,  Vol.  I,  p.  376)  that  the  theory  of  im- 
aginaries  has  now  lost  its  philosophical  importance  by  ceasing 
to  be  controversial.  There  is  still  a  hesitancy  in  the  treat- 
ment of  the  subject  in  Sixth  Forms,  which  suggests  that  in 


TOWARDS  DE  MOIVRE.    IMAGINARIES          389 

the  minds  of  at  least  some  teachers  there  is  a  lingering  doubt 
about  the  accepted  interpretation. 

Let  the  early  treatment  of  the  subject  be  frankly  dogmatic. 
Let  discussions  as  to  validity  stand  over  for  a  while. 

Define  the  symbol  V—  1  merely  as  an  expression,  (1) 
the  square  of  which  =  — -1,  and  (2)  which  follows  the  ordinary 
laws  of  algebra.  And  deduce  the  inference  that  since  the 
squares  of  all  numbers,  whether  positive  or  negative,  are 
always  positive,  it  follows  that  V— 1  cannot  represent  any 
numerical  quantity. 

Deduce  the  further  inference  that,  since  V— a2—  V—  1 X  a2 
==  V-—  1  X  #,  V—  a*  cannot  represent  any  numerical  quan- 
tity. Thus  V—  I  X  a  may  be  called  an  "  imaginary  " 
expression.  It  therefore  follows  that  such  a  statement  as 
A  +  BV—  1  =  a  +  bV  —1  can  only  be  true  when  A  =  a 
and  B  =  b.  _ 

Numbers  like  a  +  bV—  1,  where  a  and  b  are  real  numbers, 
which  consist  of  a  real  number  and  an  imaginary  number 
added  together,  are  called  complex  numbers. 

At  this  stage  it  is  advisable  to  revert  to  the  significance 
of  ordinary  negative  quantities.  If  +a  indicates  a  certain 
number  of  linear  units  in  some  chosen  direction,  —a  indicates 
the  same  number  of  linear  units  in  the  same  line  but  in  the 
opposite  direction.  Hence  when  working  out,  with  algebraic 
symbols,  a  problem  concerning  distance,  we  interpret  the 
minus  symbol  to  mean  a  complete  reversal  of  direction. 

It  is  desirable  to  take  some  little  trouble  to  convince  the 
pupils  that,  on  the  face  of  things,  there  is  nothing  in  the 
expression  a  +  b\/—l  to  make  it  more  "  absurd  "  than  in 
an  expression  like  —x.  The  result  symbolized  by  b  —  a 
where  b  is  less  than  a  is  certainly  "  imaginary  ",  unless  we 
add  to  the  conception  of  magnitude,  which  necessarily 
belongs  to  it  as  a  number,  the  further  conception  of  direction. 

Quantities  which  contain  V— 1  as  a  factor  are  obviously 
in  some  ways  very  different  from  quantities  which  do  not 
contain  it. 


390  CRAFTSMANSHIP  IN  MATHEMATICS 

What  interpretation,  then,  can  be  given  to  the  result  of 
multiplying  a  distance  by  V—  1?  Argand  put  forward  an 
ingenious  hypothesis,  which  has  now  received  general 
acceptance. 

As  we  have  seen,  the  effect  of  multiplying  a  distance  by 

—  1  is  to  turn  the  distance  through  two  right  angles. 

Hence,  whatever  interpretation  we  give  to  V— 1,  it 
must  be  such  that  the  multiplication  of  a  distance  by 

V^  X  V^-l, 

i.e.  by  —  1,  must  have  the  effect  of  turning  a  distance  through 

two  right  angles. 

Thus  it  seems  worth  while  to  consider  how  far  we  may 

interpret  the  effect  of  multiplying  a  distance  by  V—  1,  by 

supposing  that  it  turns  the  distance 
through  one  right  angle.  Evidently 
we  have  to  devise  some  scheme  by 
which  a  reversal  of  direction  will 
be  effected  in  two  identical  opera- 
tions. 

One  possible  plan  is  to  revolve 
OH  through  a  right  angle  either  in 
the  direction  of  S  or  in  the  direc- 
tion of  T,  for  each  of  these  opera- 
tions, if  repeated,  would  bring  H 

into  coincidence  with  K.      Further  double  applications  of 

the  same  operation  would  successively  bring  the  point  to 

H(+l),  to  K(— 1),  to  H  again,  and  so  on  indefinitely. 

Clearly  the  two  algebraic  operations  which,  by  definition, 

must  produce,  when  applied  in  this  way,  the  sequence  +1, 

—  1,    +1,    —  1,    ...,   represent   a   repeated   multiplication, 
either  by  +V—  1  or  by  —  V—  1. 

Thus  for  exactly  the  same  reason  that  we  identify  -—1 
with  a  unit  step  taken  along  a  line  in  a  reverse  direction  to 
the  unit  represented  by  +1,  we  may  identify  +V— 1  with 
the  revolution  of  a  line  through  a  right  angle  in  one  sense, 


TOWARDS  DE  MOIVRE.    IMAGINARIES 


391 


and  —  V— 1  with  an  equal  revolution  in  the  opposite  sense. 

This  is  the  accepted  interpretation. 

OS  is  regarded  as  the  /  (or  V— 1)  direction,  and  OT  as 
the  —i  (or  —  V—  1)  direction. 


Complex  Numbers 

Revise  the  early  work  on  the  significance  of  co-ordinates. 
— Given  a  fixed  line  OA,  and  a  fixed  origin  as  at  O,  there 
are  two  convenient  ways  of  fixing  the  position  of  a  point  P. 

1.  Rectangular  co-ordinates:   Op  —  5,  Pp  =  2. 

2.  Polar  co-ordinates:     Z.AOP  =  22°,  OP  —  5-4. 
Evidently  we  may  regard  the  rectangular  co-ordinates  as 

specifying  not  merely  measurements  which  define  the  position 

of   P,   but    also    move-  n 

ments  by  which  P  could 

be  reached  from  O.  The 

two   movements   would 

be,   one    of    +5    along 

OA  and  one  of  +2  at 


right     angles     to    OA.  . 

The  polar  co-ordinates 

specify  much  the  same  thing,  though  in  a  different  way.  If 
to  begin  with  we  are  at  O  and  facing  A,  then  the  polar  co- 
ordinates may  be  taken  as  instructions,  first  to  turn  through 
an  angle  of  22°,  and  then  advance  along  OP  a  distance  of  5-4. 

If  along  a  straight  line  a  point  takes  two  successive  move- 
ments OA,  AB,  the   length  and 
direction  of  OB   is  the  algebraic 
sum  of  the  two  movements. 

If  OA  and  OB  are  straight 
lines  or  vectors  which  represent 
two  movements  not  in  the  same 
straight  line,  the  directed  line  OB 
which  closes  the  triangle  OAB 
may  again  be  called  the  "  sum  "  of  OA  and  OB,  since  it 


B 


Fig.  236 


392  CRAFTSMANSHIP  IN  MATHEMATICS 

represents  the  single  movement  equivalent  to  the  combin- 
ation of  the  two  movements.  Thus  in  fig.  225,  OP  may  be 
called  the  sum  (more  fully,  the  vector  sum)  of  the  movements 
+5  along  OA  and  +2  at  right  angles  to  OA.  But  if  the 
movement  OP  be  represented  by  the  symbol  R,  we  cannot 
in  this  case  write  R  =  (+5)  +  (+2),  for  this  would  repre- 
sent a  movement  of  +7  from  O  along  the  line  OA.  But  we 

p  may  still  represent  R  as 

a  sum,  provided  we  do 
•/ •    \  something    to    indicate 

that  the  component  move- 
ments are  at  right  angles. 

Ui ,5 [p  A      F°r    this    purpose    the 

Fig.  227  letter   *   is    prefixed    to 

that    directed     number 

which  represents  the  component  at  right  angles  to  the  initial 
line.  This  is  in  accordance  with  our  interpretation  of  A/ — 1. 
Thus  the  movement  of  OP  would  be  represented  by  the 
notation  (+5)  +  i(+2).  (Fig.  227.) 

Of  course,  if  P  is  confined  to  the  line  OA,  a  single  directed 
number  will  suffice  to  define  its  position  after  a  series  of 
movements.  But  if  P  is  forced  to  move  about  over  the  whole 
plane  of  the  paper,  its  position  may  be  fixed  just  as  definitely 
by  such  an  expression  as,  say,  (—13)  +  *'(+21). 

Thus  we  may  regard  an  expression  of  the  form  a  +  ib 
as  a  complex  number  which  serves  to  fix  the  position  of  a 
point  in  a  plane,  just  as  the  simple  number  a  or  b  fixes  its 
position  in  a  straight  line. 

But  bear  in  mind  that  the  term  "  complex  number  "  is 
only  a  convenient  label,  suggested  by  analogy;  a  +  ib  is 
not  really  one  number  but  a  combination  of  two  numbers, 
together  with  a  symbol  i  which  stands  for  no  number  at 
all.  The  symbol  i  is  merely  a  direction  indicator — to 
show  that  the  movement  or  measurement  represented  by  the 
second  number  of  the  complex  number  is  at  right  angles 
to  that  represented  by  the  first. 

Let  a,  b  be  the  rectangular  co-ordinates,  and  r,  a  the 


TOWARDS  DE  MOIVRE.    IMAGINARIES 


393 


a 

Fig.  228 


polar  co-ordinates  of  a  point  P.  Then,  since  a  =  r  cosa 
and  b  =  r  sina, 

a  +  ib  =  r  cosa  +  i(r  sina). 

Again,  let  P'  be  the  point  on 

OP   at   unit   distance   from   O. 

Then   the  movement  OP'  may 

be  represented  by  the  complex 

number,    cosa  +  /  sina.        But 

since  r  steps,  each  of  length  OP',  would  carry  a  point  from 

O  to  P,  we  may  write: 

a  +  ib  =  (cosa  +  /sina)  X  r, 

or,  more  conveniently, 

a  +  ib  =  r(cosa  +  i  sina). 

It  follows  that  we  may  write: 

(r  cosa)  -f  i(r  sina)  =  y(cosa  -J-  i  sina). 

The  conclusion  is  important,  for  it  shows  that  we  may,  at 
least  in  this  connexion,  proceed  just  as  if  i  stood  for  a  number, 
Otherwise  we  could  not  legitimately  assume  that  the  twc 
expressions  are  equivalent. 

Note  that  the  non-directed  number  r  is  called  the  modulus 
of  the  complex  number  a  +  ib,  and  the  angle  a  its  amplitude 

The  operation  which  carries  OA  from  its  original  position 
to  OB,  then  to  OC,  then  to  OP,  in  equal  jumps,  may  be 
looked  upon  as  the  repetition  of  a  con-  _  p 

slant  factor,  viz.  a  factor  of  the  form 


cosa  +  V^OL  sina,     i.e.    cosa  -f  /  sina, 

where  a  is  the  constant  angle  between 
the  rays  from  O.  Since  two  rays 
divide  the  /.AOP  into  three  equal 
parts,  we  may  infer  that  m  —  1  rays 
would  divide  it  into  m  equal  parts. 


Fig.  229 


394  CRAFTSMANSHIP  IN  MATHEMATICS 

Hence,  if  ZAOP  =  0,0  =  ma.    Since  OA  =  r,  the  line  OP 
may  be  represented  by  the  expression: 

r(cosO  -J-  i  sinO) 

(or,  by  a  +  $,  where  a  =  r  cos#  and  b  =  r  sin#). 

Again,  since  the  factor  cosa  +  i  sina  represents  the  turning 
of  the  line  from  its  original  position  OA  through  the  angle 
a,  the  factor 

(cosa  -j-  tsina)  x  (cos(3  +  tsinp) 


must,   presumably,   represent   a  turning  through   the  angle 
(a  +  j8),  and  therefore  be  equivalent  to  the  factor 

cos  (a  +  P)  +  i  sin  (a  +  P). 

Obviously,  then,  the  identity 

coswa  -j-  i  sin;wa  —  (cosa  +  tsina)w 

is  foreshadowed.  —  The  usual  sequel  is  obvious  and  simple. 

Practice  in  the  addition  and  subtraction  of  complex 
numbers  is  desirable;  it  is  quite  easy.  Devise  examples  to 
enforce  the  notion  that  i  is  just  a  direction  indicator,  pro- 
viding us  with  a  simple  means  of  fixing  a  point  P  anywhere 
in  a  plane  containing  an  initial  line;  that  it  serves  to  show 
that  the  second  element  of  a  complex  number  is  at  right 
angles  to  the  first.  Practice  in  multiplication  and  division 
should  follow;  this  is  also  quite  easy,  once  the  boys  see  that 
cosa  +  i  sina  is  merely  a  "  direction  coefficient  ",  i.e.  a 
complex  number  which,  when  it  multiplies  another  number, 
produces  a  result  which  corresponds  to  the  turning  of  a  line 
through  the  angle  a.  De  Moivre  easily  follows. 

The  term  "  imaginary  number  "  is  not  a  happy  one; 
V—  1  is  just  a  symbol  which  can  be  treated  in  certain  cases 
as  if  it  were  a  number.  In  the  complex  number  a  +  16, 
a  is  often  called  the  real,  and  ib  the  imaginary  part. 

The  fruitful  suggestion  was  made  by  Gauss  that  instead 
of  calling  +1,  —  1,  and  V—  1,  positive,  negative,  and 
imaginary  units,  we  should  call  them  direct,  inverse,  and 


TOWARDS  DE  MOIVRE.    IMAGINARIES          395 

lateral  units.  To  Gauss  the  radical  difference  between  a 
complex  number  and  a  rational  number  was  that  while  the 
latter  denotes  the  position  of  points  along  a  line,  the  former 
denotes  the  position  of  points  in  a  plane. 

a  -f-  AV — 1  must  be  regarded  as  the  typical  number  of 
algebra,  "  real  "  numbers  being  merely  special  cases  in 
which  b  =  0.  If  we  are  confined  to  real  values  of  the  variables 
in  y  =  f(x),  we  must  admit  that  in  the  case  of  most  functions 
there  are  either  values  of  x  to  which  no  values  of  y  correspond, 
or  values  of  y  which  are  not  produced  by  any  value  of  x. 
But  if  the  variables  are  complex  numbers,  these  exceptions 
never  occur.  To  a  value  of  x  of  the  form  a  +  b\/ — 1,  there 
corresponds,  in  the  case  of  every  possible  function,  a  value 
of  y  of  the  form  A  +  BV— 1,  #>  b,  A,  B  being  themselves 
real  numbers. 

The  principle  is  so  important  that  it  must  be  understood 
thoroughly  by  all  pupils.  Emphasize  strongly  the  fact  that 
real  numbers  correspond  to  points  in  a  straight  line,  complex 
numbers  to  points  in  a  plane. — If  we  represent  the  values 
of  x  by  points  in  one  line,  and  those  of  y  by  points  in  another, 
we  cannot  say  that  any  function  y  —  f(x)  establishes  a  one- 
to-one  correspondence  between  all  the  points  on  the  two 
lines;  in  most  cases,  whole  stretches  of  points  will  remain 
outside  the  correspondence.  But  if  we  take  two  planes, 
and  represent  the  values  of  x  by  the  points  of  one  of  them, 
and  the  values  of  y  by  points  of  the  other,  we  then  obtain, 
in  every  function,  a  one-to-one  correspondence  between  all 
the  points  in  the  two  planes.  This  is  the  key  to  the  secret 
of  quadratic  equations  with  "  imaginary  "  roots. 


Quadratic  Equations  and  (so-called)  Imaginary 

Roots 

Complex  numbers  can  be  used  to  explain  certain  diffi- 
culties met  with  in  the  study  of  quadratic  equations.  Consider 
the  example  x2  —  Gx  +  34  =  0;  the  roots  of  which  are 


396 


CRAFTSMANSHIP  IN  MATHEMATICS 


sometimes  said  to  be  3  ±  V— 25.  But  x2  —  &x  +  34, 
i.e.  (x  —  3)2  +  52,  cannot  be  factorized;  hence  (we  usually 
argue)  there  is  no  value  of  x  for  which  y  (in  y  =  x2  —  6# 
+  34)  is  zero;  in  other  words,  the  equation  has  no  real 
roots.  Another  way  of  stating  this  is  that  the  parabola 
y  =  x2  —  6x  +  34  has  no  points  below  y  =  25  and  there- 
fore does  not  cross  the  axis  of  x.  Here  is  a  graph  of  the 
function: 


(-2 

\ 

70 

/ 

,6 

,\ 

/ 

\ 

\ 

_££> 

i40 

/ 

/ 

( 

),3< 

i\ 

30 

\ 

J 

29 

)- 

— 

X 

^^  \u~*> 

y 

?0 

(3,2 

^ 

10 

- 

- 

-5-3              O             ^-    +5                     +IO 

i 

Fig.  230 

#- 


34 


X    - 

-3 

O 

+  3 

+  5 

•HO 

re*  - 

9 

O 

9 

25 

IOO 

-GX- 

id 

O 

-18 

-30 

-60 

34  - 

34 

34 

34 

34* 

34 

Y  - 

61 

34 

25 

29 

74 

X 


But  if  i  be  treated  as  a  number  whose  square  is  —  1,  we  may 
write, 

(x  -  3)a  +  52  =  (x  -  3)2  -  f* .  52 

=  («?  -  3  -f-  5/)  (*  -  3  -  50. 

Apparently,  then,  y  =  0  if  jc  =  +3  ±  5i. 

It  is  usual  to  say  that  these  values  are  "  imaginary  roots  " 
of  the  equation,  or  that  they  describe  imaginary  points  where 
the  parabola  may  be  supposed  to  cross  the  axis  of  x. 

But  from  what  we  have  already  said  about  the  nature  of 
i,  there  is  clearly  an  alternative  way  of  regarding  this,  a  way 
much  more  rational.  The  values  +3  ±  5i  describe  points 
not  on  the  axis  of  x,  but  elsewhere  in  some  plane  containing 


TOWARDS  DE  MOIVRE.    IMAGINARIES         397 

that  line.  It  is  obvious  that  it  cannot  be  the  plane  of  the 
paper,  and  we  must  therefore  look  for  points  in  the  plane 
which  is  at  right  angles  to  the  plane  of  the  paper. 

The  necessary  figure  (231)  consists  of  two  parabolas,  each 
y  =  x2  —  6#  +  34,  head  to  head,  with  a  common  axis  but 
in  two  planes  at  right  angles  to  each  other.  A  suitable  sketch 
is  a  little  difficult  to  make,  but  it  may  be  done  in  this  way. — 
Let  ABCD,  EFGH  be  a  rectangular  block  with  square  ends. 
Bisect  the  block  by  the  mid-perpendicular  planes  JKLM, 
NPQR,  STUV.  The  first  and  second  intersect  in  the  line 
ab,  of  which  V  is  the  mid-point.  In  the  horizontal  plane 
fliLM,  draw  the  parabola  y  =  x*  —  fix  +  34,  with  vertex 
at  V.  In  the  vertical  plane  o/TS,  draw  the  same  parabola, 
also  with  its  vertex  at  V.  The  line  mVn  is  the  common  axis 
of  both  parabolas.  The  heavy  lines  in  the  plane  JKLM 
(xQx  and  Oy)  are  the  co-ordinate  axes  of  the  primary  parabola 
in  the  horizontal  plane.  The  axis  of  the  parabola  intersects 
the  x  axis  in  z.  As  in  the  previous  figure,  Oz  =  3,  zV  —  25. 

If,  instead  of  y  =  (x  —  3)2  +  52  the  parabola  was 
y  =  (#  —  3)2,  the  parabola  would  touch  the  axis  of  x  at  z 
(=  +  3),  but  when  the  parabola  moves  into  the  position 
y  =  (x  —  3)2  _j_  52^  its  vertex  is  at  V,  (5)2  units  from  the 
x  axis.  Hence,  the  points  given  by  the  complex  values  of 
x  answering  to  y  —  0  are  at  a  distance  5  above  and  below  the 
plane  of  the  primary  parabola,  and  on  a  similar  parabola  to 
the  first,  viz.  the  parabola  in  the  vertical  plane.  Evidently 
the  points  are  on  a  line  through  z,  m' ,  and  n'y  each  5  units 
from  z. 

Thus,  when  we  take  into  account  complex  values  of 
x,  the  complete  graph  corresponding  to  real  values  of  the 
function  y  —  (x  —  3)2  +  52  is  not  one  parabola  but  two, 
lying  in  two  planes.  The  parabola  in  the  perpendicular  plane 
contains  all  points  answering  to  complex  values  of  x  which 
satisfy  the  given  relation.  Figs.  230  and  231  should  be 
compared. 

Note  that  the  line  y  =  25  lies  in  the  plane  NPQR,  which 
is  tangential  to  both  parabolas. 


398  CRAFTSMANSHIP  IN  MATHEMATICS 


TOWARDS  DE  MOIVRE.    IMAGINARIES         399 

Note  also  the  difference  between  these  two  equations: 


*2  -  6*  -  16  =  0. 
x2  —  6x  +  9  =  25, 


3  +  5. 


*2  -  6*  +  34  =  0. 
/.   x2  -  6x  +  9  =  -25, 
/.  (96  -  3)2  =  +  V  -25  =  +  16, 
x  =  3  +  *5. 


(V25) 


Fig.  233 


To  obtain  the  points  on  the 
curve  we  proceed  from  the  origin 
to  z,  -f-3  units  away,  in  the  x  axis, 
and  then,  also  in  the  x  axis,  we 
proceed  from  z,  +6  and  —5  units, 
and  so  reach  the  points  +8  and 
—  2.  The  vertex  is  25  units  below 
the  x  axis  (see  fig.  above). 


The  journey  is  a  journey  in 
one  line,  the  x  axis.  The  two 
5's  are  measured  from  z. 


To  obtain  the  points  on  the 
curve,  we  proceed  from  the  origin 
to  s,  -|-3  units  away,  in  the  x  axis, 
and  then  we  proceed  +i5  and 
—  i5  units  from  ar,  i.e.  +5  and  —5 
units  in  a  plane  perpendicular  to 
the  plane  of  the  parabola,  where 
we  reach  points  on  a  similar  para- 
bola in  this  new  plane. 

The  vertex  of  the  primary 
parabola  is  25  units  above  the  x 
axis  (see  fig.  above  and  fig.  231). 

The  journey  is  a  journey  in 
two  lines  perpendicular  to  each 
other.  The  two  5's  are  measured 
from  z  as  before,  but  in  a  perpen- 
dicular plane. 


400  CRAFTSMANSHIP  IN  MATHEMATICS 

Unless  provided  with  a  wire  model  (fig.  231),  or  with  a 
really  good  perspective  sketch,  boys  are  apt  to  be  puzzled 
by  this  problem.  A  model  is  much  to  be  preferred;  then 
the  effect  of  increasing  and  decreasing  the  distance  of  the 
primary  parabola  from  the  x  axis  is  easily  observed. 

Warn  the  boys  not  to  be  led  away  by  the  remarkable 
and  perfectly  logical  consistency  of  the  hypothesis  concerning 
V — 1.  It  is  only  an  hypothesis  after  all.  Still,  it  is  not 
advisable  for  learners  to  talk  about  "  imaginary  "  roots  of 
equations  but  rather  to  explain  such  roots  in  the  light  of 
the  hypothesis  in  question. 

We  have  touched  upon  vector  algebra.  The  subject 
receives  considerable  attention  in  Technical  Schools  but 
very  little  in  Secondary.  This  is  a  pity,  for  it  is  a  cunningly 
wrought  instrument  and  is  as  useful  as  it  is  illuminating. 
Quite  the  best  introduction  to  it  is  Part  I  (Kinematic)  of 
Clifford's  Elements  of  Dynamic.  The  first  two  parts  of  the 
book,  Steps,  and  Rotation,  should  be  read  by  all  teachers  of 
mathematics,  and  the  third  part,  Strains,  by  all  teachers  of 
mechanics.  Maxwell's  Matter  and  Motion  is  a  little  book 
dealing  admirably  with  the  same  subject.  Henrici  and 
Turner's  Vectors  and  Rotors  is  also  useful. 


TOWARDS  THE  CALCULUS  401 

CHAPTER  XXIX 

Towards   the   Calculus 

Co-ordinate  Geometry 

Teachers  differ  in  opinion  whether  the  calculus  should 
be  preceded  by  a  course  of  co-ordinate  geometry.  Certainly 
anything  like  a  complete  course  of  co-ordinate  geometry  is 
not  a  necessary  preliminary.  On  the  other  hand,  some  little 
knowledge  of  its  fundamentals  is  advisable,  and  this  is  easily 
developed  from  the  previous  knowledge  of  graphs.  The 
notion  of  the  differential  coefficient  is  nearly  always  made 
to  emerge  from  considerations  of  the  tangent  to  the  parabola, 
but,  more  frequently  than  not,  the  common  properties  of 
the  parabola  have  not  previously  been  taught.  This  partly 
explains  the  haze  which  often  enshrouds  the  notions  under- 
lying the  new  subject. 

A  minimum  of  preliminary  work  in  co-ordinate  geometry 
may  be  outlined. 

The  boys  already  know  that  y  —  mx  +  c  represents  a  line 
making  an  angle  whose  tangent  is  m  with  the  axis  of  x\  that,  in 
short,  m  represents  the  slope  of  the  line;  and  that,  in  whatever 
other  form  the  equation  may  be  written,  it  may  be  re-cast 
into  the  y  —  mx  form,  and  its  slope  be  determined  at  once. 

X  *V 

For  instance,  the  intercept  form  —  ~  +  ~  =  1  mav  be  written 

3 


y  =  —  ^/%x  -f  3,  and  the  slope  is  seen  to  be  —  •  \/3. 

The  boys  must  be  able  to  determine  the  equation  of  a 
line  satisfying  necessary  conditions.  They  already  know  that 
if  they  are  told  a  straight  line  must  pass  through  a  given 
point,  this  condition  alone  is  not  enough  to  determine  the 
line,  since  any  number  of  lines  may  pass  through  the  point. 
But  if  they  are  given  some  second  condition,  e.g.  the  direction 
of  the  line,  or  the  position  of  a  second  point  through  which 

(£291  27 


402 


CRAFTSMANSHIP  IN  MATHEMATICS 


it  passes,  then  the  two  data  completely  fix  the  line.  This 
fits  in  with  the  fact  that  the  equation  of  the  line  must  contain 
two  constants. — Make  the  boys  thoroughly  familiar  with  the 
ordinary  rules  for  finding  the  equation  of  a  line  satisfying 
two  given  conditions. 

Types  of  suitable  exercises  for  blackboard  oral  work: 

1.  Find  the   equation   of  a  straight  line  cutting  off  an 
intercept  of  2  units'  length  on  the  axis  of  y  and  passing  through 
the  point  (3,  5). 

2.  Find  the  equation  of  a  straight  line  drawn  through 
the  point  (3,  5),  making  an  angle  of  60°  with  the  axis  of  x. 

3.  Find  the  equation  of  a  straight  line  passing  through 
the  points  (2,  3),  (-4,  1). 

The  last  exercise  is  a  type  with  which  the  boys  should  be 
thoroughly  familiar.  It  will  be  required  often  in  future 
work.  The  equation  should  therefore  be  familiar  in  its 
general  form,  and  should  be  illustrated  geometrically. 

Find  the  equation  of  a  line  passing  through  the  two  points 
A(#i,  jVi)  and  B(#2,  y2).  We  may  proceed  in  this  way: 


W-T 


Suppose   y  =  mx 
m  and  c  are  unknown. 


c    represents    the    equation,    where 


The  particular  point  (#t,  yx)  is  on  the  line,  .'.  yl  —  mxl  +  c.  (i) 
The  particular  point  (x2,  y2)  is  on  the  line,  .'.  y%  =  mx2  -f-  c.  (ii) 
The  point  (#,  y),  any  point,  is  on  the  line,  .'.  y  =  mx  +  c.  (iii) 


TOWARDS  THE  CALCULUS  403 

Subtracting  (i)  from  (iii), 

y  —  y1  =  m(x  —  xj. 

Subtracting  (i)  from  (ii), 

^2  —  y\  =  "K*2  —  *i)- 
.'.  by  division 


or        (y  -  yi)  - 


which    is    the    required    equation,   -^  -  ~    representing    m 
(the  slope)  in  y  =  mjc.  ^2  ~  ^ 

Beginners  rarely  see  this  clearly,  unless  the  algebra  is 
clearly    illustrated     geometrically.      From    the    last    figure, 


take  out  the  two  similar  triangles,  and  show  the  lengths 
of  their  perpendiculars  and  bases  in  terms  of  co-ordinates. 
The  slope  of  the  line  AB  is  given  from  the  first  triangle; 

=  9* i-J,  A  and  B  being  the  two  specified  points  on 

AK      #2  ~"  #1 
the  line. 

The  slope  of  the  line  AB  is  also  given  from  the  second 

triangle,  — -  =  y  ~~  ?\  A   being  a  specified  point,  and   P 
AM       x  —  #! 

being  any  point  on  the  line. 


404  CRAFTSMANSHIP  IN  MATHEMATICS 

But  the  triangles  are  similar,  and  the  slopes  are  therefore 
identical. 

.-  y  ~~  yi  =  yLny\ 

X  —  Xl          X2  —  Xj* 

or        (y  —  yx)  =  ^-"H^1  (x  —  x^,  as  before. 

#2          Xi 

Or  again:   take  the  slope  found  in  the  last  case,  say  —  —  —  , 
and  substitute  for  m  in  y  ~  mx  +  c.    We  have  *2  ~~  x* 


a) 


and  since  the  line  passes  through  (xly  j/x),  [or  (#2,  y2)  might 
be  chosen  if  preferred], 


By  subtraction,  we  have  from  (i)  and  (ii) 

~  x\)t  as  before. 


Every  step  must  be  substantiated  geometrically.  It  is 
fatal  to  allow  the  boys  to  look  upon  the  problem  as  mere 
algebra.  The  boys  should  be  able  to  write  down  instantly  the 
equation  of  a  line  passing  through  two  points  and  under- 
stand its  full  significance.  But  in  evaluations  of  this  kind, 
do  not  be  satisfied  with  just  the  typical  textbook  formal 
solutions.  Vary  the  work. 


The  Parabola  and  its  Properties 

Throughout  the  teaching  of  co-ordinate  geometry,  let  all 
principles  be  established  first  by  pure  geometry.  Let  the 
picture  come  first,  and  teach  its  new  lessons.  Then  let 
symbolism  follow.  Geometry  treated  as  pure  algebra  tends 
to  lose  all  semblance  of  its  essential  space  relations.  At 


TOWARDS  THE   CALCULUS 


105 


least  the  parabola,  if  not  the  ellipse,  will  already  have  received 
some  attention.  It  will  have  been  touched  upon  in  connexion 
with  graphs  and  quadratic  equations,  and  the  boys  may  have 
learnt  something  about  the  paths  of  falling  bodies;  they 
may  also  know  that  in  certain  circumstances  the  chains  of  a 
suspension  bridge,  and  vertical  sections  of  the  surface  of  a 
rotating  liquid,  form  parabolas. 

It  is  useful  to  give  the  boys  a  mechanical  means  of  readily 
drawing  a  parabola.  It  saves  much  time,  and  encourages 
them  to  use  good  figures.  Here  is  one  way. 

An  ordinary  T-square  slides  along  AB,  the  left-hand  edge 
of  a  drawing  board,  in  the  usual  way,  the  edge  AB  answering 


C 


Fig.  235 


as  a  directrix.  A  string  equal  in  length  to  KG  is  fastened 
at  G,  and  at  a  fixed  point  S  in  a  line  XX  perpendicular  to 
AB.  A  pencil  point  P  keeps  the  string  stretched  and  remains 
in  contact  with  the  edge  KG  of  the  T-square.  As  the  T-square 
slides  up  and  down  the  edge  of  the  board,  the  pencil  traces 
out  a  parabola  with  focus  S. 

A  parabola  is  the  locus  of  a  point  whose  distance  from  a 


406 


CRAFTSMANSHIP  IN  MATHEMATICS 


fixed  point  is  equal  to  its  distance  from  a  fixed  straight 
line. — Help  the  boys  to  see  that  from  this  definition  certain 
properties  follow  at  once: 

(1)  ±  PM  (diam.)  =  PS. 

(2)  ES  (semi-latus  rectum)  =  EM'  =  SX. 

(3)  AS  =  AX. 

(4)  SX  =  2SA. 

(5)  ES  -  2SA. 

(6)  EF  (latus  rectum)  =  4SA. 


The  main  property  to  be  mastered  is  the  slope  of  the 
tangent,  and  to  this  end  the  following  summary  of  pre- 
liminary work  is  suggested.  All  principles  should  be  established 

first  geometrically,  then 

0  -  analytically,     and      the 

boys  must  be  made  to 
see  that  the  results  are 
identical. 

1 .  The  principal 
ordinate  of  any  point  P 
on  a  parabola  is  a  mean 
proportional  to  its  ab- 
scissa and  the  latus  rec- 
tum. 

i.e.  PN2-  4AS.AN. 

Analytically:  call  the 
point  P,  (x9  y)\    SA  = 
Fig.  236  AX  =  *(8ay).    Theny2 

-  4ax.    (Fig.  236.) 

If  the  directrix  MX  is  the  y  axis,  the  equation  becomes 
y2  =  &a(x  —  a). 

2.  If  a  chord  PQ  intersects  the  directrix  in  R,  SR  bisects 
the  external  angle  QSP'  of  the  triangle  PSQ. 


TOWARDS  THE  CALCULUS 


407 


Drop  JLs  PM 
and  QV  on  direo 
trix.  As  PMR  and 
QVR  are  similar. 


PR 
QR 


PM 
QV 


PS 
QS' 


Fig.  237 


3.  If  the  tangent 
at  P  meets  the  direc- 
trix in  R,  the  angle 
PSR  is  a  right  angle. 
— Deduce  this  from 
the   preceding    pro- 
position.— When    Q 
coalesces     with     P, 
each  of  the  marked 

equal  angles  becomes  a  right  angle.     (See  next  figure.) 

4.  The  tangents  at  the  extremities  of  a  focal  chord  intersect 
D 


PP«  jbcal  chord 
RP,  RP  -  tangents 
Z.PRP*- 


Fig.  238 


at  right  angles  on  the  directrix,  i.e.  the  tangents  at  P  and  P', 
the  extremities  of  the  focal  chord  PSP;,  make  a  right  angle 


408 


CRAFTSMANSHIP  IN  MATHEMATICS 


at  R,  where  they  meet  on  the  directrix.     Observe  that  RS 
meets  the  focal  chord  at  right  angles  (cf.  No.  3). 

5.   The  subtangent  is  equal  to  twice  the  abscissa,  i.e.  TA= AN 

or,  TN  =  2AN. 


Fig.  239 


6.  The  foot  of  the  focal  perpendicular  of  any  tangent  lies 
on  the  tangent  at  the  vertex,  i.e.  Y,  the  foot  of  the  -L  SY  to 
the  tangent  PT,  lies  on  the  tangent  at  A. 


Fig.  340 


TOWARDS  THE  CALCULUS  409 

»    rr>i      7         r          .         .2  latus  rectum      ~ 
7.  1  he  slope  of  any  tangent  =  -  -  -  -  -.     ror  m 
^      J       J        6  ordinate 

the  last  figure  the  triangles  YAS  and  TNP  are  similar.  Hence 
slope  of  tangent  PT 

^  PN  =  SA  =  2AS  ^  \  latus  rectum 

""  NT  ~  AY  ~~  PN  ordinate      ' 

i.e.  the  slope  of  the  tangent  to  the  axis  of  the  parabola.     If 
the  figure  is  turned  round,  so  that  the  slope  of  PT  is  to  the 

ATT          i  11  ordinate 

tangent   AY   at   the   vertex,   then   slope  =  —  -  --  . 

$  latus   rectum 

How   may   this  slope  be  expressed   in   rectangular   co- 
ordinates?    The  equation  of  a  secant  cutting  the  curve  in 

p*>      and  Q          is 


If  a  figure  be  drawn  accurately  (this  is  not  easy),  actual 
measurement  will  show  that  the  slope  of  the  secant  is  the 
ratio 

\  latus  rectum^ 
mean  of  ordinates 

To  obtain  the  equation  of  the  tangent  at  (xlt  y^  we  take 
Q  indefinitely  close  to  P,  so  that  ultimately  j>2  =  yv  The 
equation  to  the  secant  then  becomes: 


a,      , 
or    y  =  —  (x  +  *! 


which  is  the  equation  to  the  tangent,  and  the  slope  of  the 

*  •    *u      20   .      4  latus  rectum        u  - 
tangent  is  thus  —  ,  i.e.  =  -  -  -  ,  as  before. 
yl  ordinate 


410  CRAFTSMANSHIP  IN  MATHEMATICS 

The  Tangent  to  the  Parabola 

If  future  work  is  to  be  understood,  the  tangent  to  the 
parabola,  and  its  various  implications,  must  receive  close 
attention.  The  necessary  further  elucidation  may  thus  be 
summarized. 

1.  To  find  the  condition  that  the  straight  line  y  =  mx  -f-  c 
may  touch  the  parabola  y2*  —  kax. 

Since  y  =  mx  -f-  c, 

.'.  y2  -  (mx  +  c)*; 
and  since  y2  —  lax, 
:.  (mx  +  f)2  =  lax. 

By  solving  this  equation  we  obtain  the  abscissae  of  the  two 
points  in  which  the  straight  line  cuts  the  curve.  The  line 
will  touch  the  curve  if  the  two  points  coincide,  and  the  con- 
dition for  this  is  that  the  roots  must  be  equal, 

i.e.  in  m2*2  +  2x(mc  -  2a)  +  c2  =  0, 

l(mc  -  2a)2  =  Im2c29 

a 

i.e.    a  =  me,    or    c  —  — . 
m 

Hence  the  line  y  =  mx  +  c  touches  the  curve  y2  —  kax  if 

c  =  _  (where  m  is  the  slope  which  the  tangent  makes  with 
m 

the  axis). 

2.  To  find  the  point  where  the  tangent  y  =  mx  +  —  touches 
the  parabola  y  —  4a#. 

As  before, 

(mx  +  c)2  =  lax, 

/.   [mx  -j-  — )    ==  40#, 
>•  m' 


( 


or     \  mx 1    = 


and  since  y2  =  lax,    y  =  — . 


TOWARDS  THE  CALCULUS  411 

a    2a\ 


Hence  the  point  required  is  ( — -,  — ). 

\m2  m/ 


3.  Compare  the  two  forms  of  the  equation  of  the  tangent, 


viz.  yy±  =  2a(x  -f-  #i)>  and  y  =  mx  +  —  . 

m 


The  first  may  be  written, 


2a 

y  =  —  x 


Hence  we  may  write  the  two  forms  in  parallel  thus: 


y  =  mx    +  - 
nt 


T»U  *  *u  r  .       a  , 

They  represent  the  same  line;     .  .   —  —  m,  and 

Ji  i 

That  these  two  last  equations  are  consistent  may  easily  be 
shown  by  evolving  the  second  from  the  first,  the  connecting 
link  being  y^  = 


<0    •       t_ 

Evidently,  then,  the  equation  y  =  mx  +  —  1S  the  tangent 

o  ^^ 

at  the  point  (xl9  y^,  i.e.  ( — ,  — ). 

\m     m  / 

4.   Verify    geometrically    that    the    tangent   y  =  w#  +  — 

9  Wl 

touches  the  parabola  y2  =  &ax  at  the  point  ( — ,   — j.     (This 
verification  is  of  great  importance.)  m 

(The  abler  boys  ought  to  do  this  without  any  further 
help). 

AY  =  tangt.  at  vertex  =  y  axis. 
AX  =  x  axis. 
S  =  focus. 

TA  =  AN  (subtangent  =  2  abscissa) 

SZ  meets  the  tangent  PT  at  rt.  Z.s  at  Z,  since  AZ  is  the  tangent  at 
vertex. 


412 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  rt.  angled  As  TAZ,  ZAS,  TZS,  TNP  are  all  similar. 
Since  TA  =  £TN,  /.  ZA  =  £PN. 
AS  =  a  =  dist.  of  focus  to  vertex. 


Fig.  241 


AZ  =  —  =  intercept  on  y  axis. 
m 

PN 

— -  =  m  —  slope  of  tangent. 

TA    _  AZ        _.   _  AZ2 
_=  _,or  I*-—, 


/.  AN 


AZ2 
AS 


a 


—    —  =  abscissa  of  P, 


and 


2a 


PN  =  2ZA  =  —  =  ordinate  of  P. 


Q.E.D. 


Observe,     again,     that      the     slope     of     the     tangent 
__  \  latus  rectum  __  2a 


ordinate 


—  —  -  =  m. 
2a 


m 


This  pictorial  parallelism  between  the  geometry  and 
algebra  is  essential  whenever  it  is  possible.  Let  the  boys 
see  that  co-ordinate  geometry  is  geometry  and  not  mere 
algebra.  But  of  course  the  geometrical  figure  is  also  a 
graph,  to  be  interpreted  algebraically. 


We   have   taken    the   subject   of   co-ordinate    geometry 


TOWARDS  THE  CALCULUS  413 

thus  far,  less  for  its  own  sake  than  as  an  introduction  to  the 
next  chapter.  Co-ordinate  geometry  is  an  easy  subject  to 
teach,  and  boys  like  it,  provided  the  geometry  itself  is  made 
clear.  As  a  subject  of  mere  algebraic  manipulation,  un- 
associated  with  pure  geometry,  its  value  is  slight,  and  time 
should  not  be  spent  over  it. 

Methods  of  Approximation 

The  calculus  is  such  a  valuable  mathematical  weapon, 
and  the  fundamental  ideas  underlying  it  are  so  simple,  that 
the  subject  should  find  a  place  in  every  Secondary  school. 
It  might  be  begun  in  the  Fifth  Form,  if  not  in  the  Fourth, 
though  naturally  the  first  presentation  must  be  of  a  simple 
character.  This  simple  presentation  is  easily  possible.  The 
more  technical  side  of  the  subject,  as  elaborated  in  the 
standard  textbooks,  is  wholly  unnecessary  in  schools. 

It  was,  I  believe,  Professor  Nunn  who  pointed  out  that 
the  history  of  the  subject  suggests  the  best  route  for  teachers 
to  follow.  Although  Newton  and  Leibniz  are  rightly  given 
the  credit  of  being  the  creators  of  the  calculus  as  a  finished 
weapon,  the  preliminary  work  of  certain  of  their  predecessors, 
especially  Wallis,  from  which  the  main  idea  of  the  calculus 
was  derived,  must  always  be  borne  in  mind.  Wallis's  work 
is  merely  a  special  kind  of  algebra  and  may  readily  be  under- 
stood by  a  well-taught  Fourth  Form. 

If  we  are  thus  to  begin  with  approximation  work,  there 
is  much  to  be  said,  as  pointed  out  by  Professor  Nunn,  for 
beginning  with  integration  rather  than  with  differentiation. 
The  necessary  arguments  are  so  simple  and  the  results 
so  valuable  that  the  rather  radical  departure  from  normal 
sequence  is  justified.  For  all  practical  purposes,  Wallis 
was  the  actual  inventor  of  the  integral  calculus,  and  Wallis 's 
own  work  and  methods  serve  to  give  young  pupils  a  clear 
insight  into  the  new  ideas. 

This  early  work,  in  differentiation  as  well  as  in  integra- 
tion, should  be  taught  as  a  calculus  of  approximations.  The 


414  CRAFTSMANSHIP  IN  MATHEMATICS 

pupils  should  learn  that  such  investigations  give  results 
which  may  be  regarded  as  true  to  any  degree  of  approxima- 
tion, though  not  absolutely  true.  When  later  the  calculus 
itself  is  formally  taken  up,  and  the  pupils  are  able  to  grasp 
the  modern  theory  of  "  limits  ",  they  should  be  able  to  see 
that  the  new  arguments,  if  properly  stated,  do  as  a  matter 
of  fact  give  results  which  are  unequivocally  exact.  They 
must  not  be  allowed  to  assume,  at  that  later  stage,  that  the 
arguments  of  the  calculus  prove  merely  approximately  true 
results,  and  yet  that  these  may  be  treated  as  if  they  were 
exact  truths.  This  illegitimate  jump  from  possible  truth 
to  certain  truth  is  often  made,  it  is  true;  but  the  deduction 
commonly  involves  the  fallacious  use  of  such  terms  as 
"  infinitely  small  ",  "  infinitely  great  ",  and  the  like. 

"  Methods  of  approximation  are  inferior  methods  and 
do  not  yield  exact  results."  Granted.  But  these  methods 
are  best  for  beginners,  if  only  because  they  form  a  good 
introduction  to  the  exact  methods  of  the  calculus,  and  they 
are,  after  all,  based  upon  a  kind  of  reasoning  which  is  rigor- 
ous enough  for  practical  purposes.  But  the  important  thing 
is  to  make  the  pupils  feel  that  they  must  never  be  finally 
satisfied  until  they  have  mastered  a  method  which  yields 
results  that  admit  of  no  doubt  at  all. 

The  beginner  has  already  learnt,  or  should  have  learnt, 
from  his  graph  work  the  main  idea  of  the  real  business  to 
be  taken  in  hand,  and  that  is  the  nature  of  a  function:  that 
the  value  of  one  variable  can  be  calculated  from  the  value 
of  another  by  the  uniform  application  of  a  definite  rule 
expressed  algebraically. 

We  append  a  few  suggestions  for  work  in  suitable  approxi- 
mations. 

1.  Rough  approximations. 

(i)  Revise  certain  exercises  in  mensuration,  e.g.  find  the 
area  of  a  circle  and  of  some  irregular  figures  by  the  squared 
paper  method. 

(ii)  Surveyor's  Field-Book  exercises;    measure  up  some 


TOWARDS  THE  CALCULUS  415 

irregular  field,  or  other  area,  but  insist  that  all  such  results 
are  only  rough  approximations. 

2.  Closer  approximations,  and  the  methods  involved. 

(i)  Revise  the  work  on  expansion  (in  physics).  For  instance, 
the  coefficient  of  linear  expansion  of  iron  is  '00001.  Justify 
the  rule  of  accepting  -00002  instead  of  (-00001)2  for  area 
expansion,  and  -00003  instead  of  (-00001)3  for  cubical  expan- 
sion. Show  the  utter  insignificance  of  the  rejected  decimal 
places.  Refer  to  the  geometrical  illustrations  of  (a  +  b)z 
and  (a  +  ft)3. 

(ii)  Estimate  the  area  of  a  triangle  as  the  sum  of  a  number 
of  parallelograms.  The  more  numerous  the  parallels  and  the 


z w          £. 

Fig.  243 

more  numerous  the  parallelograms,  the  more  negligible  do 
the  projecting  little  triangles  become.  Observe  that  if  the 
number  of  parallelograms  is  doubled,  each  shaded  triangle 
is  reduced  to  one-fourth;  and  so  on. 

(iii)  Estimate  the  volume  of  a  pyramid  as  the  sum  of  the 
volumes  of  a  number  of  flat  prisms,  gradually  diminished 
in  thickness. 

(iv)  Estimate  the  volume  of  a  sphere  regarded  as  the  sum 
of  a  number  of  pyramids  formed  by  joining  the  centre  of  the 
sphere  to  the  angular  points  of  a  polyhedron,  the  number  of 
whose  faces  is  increased  indefinitely.  The  pyramids  formed 
have  as  their  bases  the  faces  of  the  polyhedron;  the  volume 
of  each  pyramid  —  (face  X  height) /3,  hence  the  volume  of 
the  sum  of  the  pyramids  =  (sum  of  faces)  X  height/3.  If 
the  number  of  faces  be  increased,  the  sum  of  the  faces  becomes 
more  nearly  equal  to  the  area  of  the  spherical  surface,  and 
then  the  height  of  the  pyramid  is  more  nearly  equal  to  the 


416  CRAFTSMANSHIP  IN  MATHEMATICS 

radius  r  of  the  sphere.  But  the  sum  of  the  faces  can  never 
be  quite  equal  to  the  surface  of  the  sphere,  though  we  can 
so  increase  the  number  of  faces  of  the  polyhedron  that  the 
approximation  may  be  closer  than  any  degree  we  like  to 
name.  The  spherical  surface  is  necessarily  greater  than  the 
sum  of  the  flat  faces  of  the  polyhedron  and  can  never  be 
reached:  it  is  an  unreachable  limit.  If  the  sum  of  the  faces 
could  become  equal  to  the  surface  of  the  sphere  the  sum 
would  be  4:irr2  and  then  the  height  of  each  pyramid  would 
be  equal  to  r.  Hence  the  volume  of  the  sphere  would  be 

r       4 

4?rr2  X  -  =  -77T3.      Now  this   result   agrees   with   the   result 
3       o 

arrived  at  by  other  methods,  and  it  is  correct.  Still,  to  obtain 
the  result,  we  had  to  jump  from  flat-faced  pyramids  (though 
these  may  have  been  made  inconceivably  small)  to  corre- 
sponding bits  of  spherical  surface  which  were  not  flat. — 
We  have  still  to  discover  whether  such  a  method  is  allow- 
able. 

(v)  The  value  of  TT.  The  pupils  may  be  allowed  to  assume 
— from  a  figure  they  will  readily  guess — that  if  2  regular 
polygons  with  the  same  number  of  sides  be,  respectively, 
inscribed  within  and  circumscribed  without  a  circle,  the 
length  of  the  circumference  of  the  circle  will  be  less  than 
the  perimeter  of  the  circumscribed  polygon  but  greater  than 
that  of  the  inscribed  polygon.  Show  the  pupils  that  the 
determination  of  TT  is  thus  merely  a  question  of  arithmetic, 
though  of  very  laborious  arithmetic,  inasmuch  as  we  have  to 
determine  the  perimeters  of  polygons  of  a  very  large  number 
of  sides;  the  greater  the  number  of  sides,  the  greater  the 
degree  of  approximation  of  the  value  of  TT. — Give  a  short 
history  of  the  evaluation  of  TT,  from  the  time  of  Archimedes 
onwards.  Point  out  that  the  irrationality  of  TT  has  now  been 
definitely  demonstrated,  so  that  it  is  useless  for  computers 
to  waste  any  more  time  over  it.  Make  the  pupils  see  that 
the  method  of  evaluating  TT  is  only  a  method  of  approximation, 
and  that  in  this  case  no  better  method  is  ever  to  be  hoped 
for;  that  we  can  obtain  values  more  and  more  approximating 


TOWARDS  THE  CALCULUS 


417 


to  the  ratio  of  the  circumference  to  the  diameter,  but  there 
can  be  no  final  value,  as  the  decimal  can  never  terminate. 


Area  under  a  Parabola 

We  cut  off  a  parabola  by  a  line  P'P  perpendicular  to  its 
axis  OA,  and  enclose  it  in  the  rectangle  P'M'MP.  We  will 
calculate  the  shaded  area  OPM,  i.e.  the  area  "  outside  "  or 
"  under  "  the  half  parabola  AOP.  Let  the  parabola  bey=kx2 


We  may  divide  OM  into  any  number  of  equal  parts,  and 
on  these  parts  construct  a  number  of  rectangles  of  equal 
breadth,  set  side  by  side  as  shown  in  the  figure.  The  added 
areas  of  the  rectangles  are  evidently  in  excess  of  the  area 
OPA,  but  by  increasing  the  number  of  rectangles  indefinitely, 
the  excess  is  indefinitely  diminished. 

We  will  begin  by  dividing  OM  into  a  small  number  of 
parts,  and  then  increase  the  number  gradually.  As  the  first 
division  OQ  is  gradually  to  be  diminished,  we  will  consider 
the  rectangle  on  it  to  be  of  zero  area.  Hence  OR  =  (2  —  1) 
=  1  unit;  .'.  RR'  =  I2  units,  OS  =  (3  —  1)  =  2  units; 

(K291)  28 


418  CRAFTSMANSHIP  IN  MATHEMATICS 


A                                          PA 

L 

~\                     I 
TOTAL-  ARC*  AM  . 
DIV  DEO   INTO  4-32 

-as. 

— 

TOTAL  AREA  AM 

DIVIDED    INTO  3*2* 

-12 

2* 

—  -  -4                 ' 

.  .  _,  _           s 

.av. 

R 

z 

I2 

L|Z 

___  TOT 


M 

0  2  O*i25  0123* 


P  A 


TOTAL  AREA  A  M 

TOTAL  AREA  A  M 

TOTAL   AREA  A  M 

DIVIDED  INTO  6*5 

DIVIDED   INTO  7  "6 

DIVIDED    INTO  8  X7 

-  150 

•  252 

1  — 

-39E 

j 

2 

— 

6 

7 

1  

2 

2 

5 

G' 

2 

2 

I  3 

FT 

4 



, 

5 

II 

i.t 

1   * 

4 

i  12    1 

[?[ 

P]5 

D9RSTUM             09RSTUVM             O9RST 

U     V    W    M 
4567 

(IV) 


(v) 
Fig.  244 


.'.  SS'  -  22  units;    OT  =  (4  -  1)  =  3  units,   /.  TT'  = 
units;   &c. 

We  may  tabulate  the  results  thus: 


Linear 
Units  in 

Square  Units  of  Area  in 

Ratio  of 
(*)  to  (a). 

OM. 

PM. 

(a)  Rect. 
AM. 

(6)  Sum  of  contained  Rectangles. 

Lowest 
Terms. 

Re- 
written. 

3 

22 

3  x  22 

12  +  22 

A 

A 

4 

32 

4  X  32 

I2  +  22  +  32 

A 

A 

5 

42 

5  X  42 

I2  -}-  22  -f  32  +  42 

a 

8 

9 
24 

'6 

5a 

6  X  62 

p  +  22  +  32  +  42  +  52 

M 

a* 

7 

62 

7  X  62 

I2  +  22  +  32  -f  42  -f  52  +  62 

M 

11 

8 

72 

8  X  72 

I2  +  22  +  32  -f  4a  +  52  +  62  +  72 

A 

M 

TOWARDS  THE  CALCULUS  419 

The  rewritten  ratios  show  the  numerators  in  A  .P.,  and  the 
denominators   as   multiples   of  6.     Obviously  if  m  be  the 

number  of  units  in  QM,  the  ratio  may  be  written  —  --—  or 
1  Gm 


Thus  the  area  of  the  added  rectangles  is  equal  to  £  the 
area  of  the  rectangle  AM  +  a  fraction  depending  on  the 
value  of  m.  It  is  easy  to  prove  that  the  law  holds  good  for 
all  values  of  m. 

The  ratio    —  i—  _   enables  us  to  write  down  as  many 
6m  J 

terms  as  we  please.  For  instance  if  m  —  1000,  the  ratio 
=  elH)  o  or  4  +  6  oW  •  Hence  if  we  built  up  a  figure  with 
1000  rectangles,  the  total  area  of  the  rectangles  would  be  equal 
to  £  of  AM  +  a  small  area  equal  to  Q-^Q  of  AM. 


Evidently   by   taking   m   large   enough,   the   fraction  — 

6m 

becomes  so  small  as  to  be  insignificant,  and  thus  the  com- 
bined area  of  the  rectangles  can  be  made  to  differ  as  little 
as  we  please  from  |  the  area  of  AM.  And  as  the  rectangles 
are  made  narrower  and  narrower,  the  area  they  cover  will 
eventually  become  indistinguishable  from  the  area  under 
the  curve  OP  (fig.  243);  e.g.  if  m  =  1000  the  sum  of  the  top 
left-hand  corners  of  the  rectangles  projecting  outside  the  curve 
is  only  -Q-QOQ  °^  AM.  Finally  the  tops  of  the  rectangles  will 
be  indistinguishable  from  the  curve  itself.  We  conclude, 
therefore,  that  this  area  under  the  curve  is,  at  least  to  a  very 
great  degree  of  accuracy,  |  of  the  rectangle  AM.  Since 
OM  =  x,  and  PM  —  kx*  (fig,  243),  we  express  the  conclusion 
by  the  formula 

A  = 


It  follows  that  the  area  AOP  is  f  the  area  AM.    Hence  the 
whole  area  of  the  parabola  up  to  PT  is  f  OA  X  P'P. 

A  point  for  emphasis:  "  Having  proved  that  the  area 
under  the  curve  is,  apparently  to  an  unlimited  degree  of 
closeness,  \  of  the  rectangle  AM,  we  are  almost  forced  to 


420 


CRAFTSMANSHIP  IN  MATHEMATICS 


believe  that  the  former  is  exactly  \  of  the  latter."  Still,  the 
fact  remains  that  what  we  have  proved  is  only  an  approxima- 
tion. Do  not  disguise  the  theoretical  imperfection  of  the 
conclusion.  Do  not  slur  over  the  fact  that  we  have  merely 
an  approximation  formula,  though  it  is  quite  proper  to  em- 
phasize the  other  fact  that  no  limit  can  be  set  to  the  close- 
ness of  the  approximation  which  it  represents. 

The  particular  approximation  result  arrived  at  is  easily 
extended. — Let  an  ordinate  start  from  the  origin  and  move 
to  the  right.  If  it  has  a  constant  height,  y  =  k,  it  will,  in 
moving  through  a  distance  x,  trace  out  an  area,  A  =  kx. 
If  its  height  is  at  first  zero,  but  increases  in  accordance  with 
one  of  the  laws  y  =  kx,  y  =  kx*,  y  ----  kx3,  the  area  traced 
out  will  be  given  by  the  corresponding  law,  A  —  \kx*> 
A  =  J&#3,  A  =-•  \kofi.  These  results  we  might  establish  by 
proceeding  exactly  as  before.  Calling  the  function  which 
gives  the  height  of  the  ordinate,  the  ordinate  function,  and 
the  function  which  gives  the  area  traced  as  the  area  function^ 
the  results  may  be  summarized  simply 


Ordinate  functions 
Corresponding  area  functions 

kx° 

kx1 
1 

kx2 

•s«* 

kx3 
4  iX 

kxn~l 

~kxn 
n 

This  summary  exhibits  Wallis's  Law. 

Books  to  consult: 

1.  Teaching  of  Algebra,  Nunn. 

2.  Cartesian  Plane  Geometry,  Scott. 

3.  Modern  Geometry,  Durell. 


THE  CALCULUS:    SOME  FUNDAMENTALS      421 
CHAPTER  XXX 

The  Calculus :     some   Fundamentals 

First  Notions  of  Limits 

To  boys  the  two  terms  "  infinity  "  and  "  limits  "  are 
always  bothersome,  and  it  is  doubtful  if  the  first  ought  to 
be  used  in  Forms  below  the  Sixth.  A  misapprehension  as 
to  the  significance  of  both  terms  is  responsible  for  much 
faulty  work,  much  fallacious  reasoning. 

What  is  a  point?  "  A  point  is  that  which  marks  position 
but  has  no  magnitude."  But  how  can  a  thing  with  no  magni- 
tude indicate  position?  If  it  has  magnitude,  is  it  of  atomic 
dimensions,  say  10~24  cm.?  Or  is  it  10~10  of  this?  Obviously 
if  it  has  magnitude  at  all,  a  certain  definite  number  side  by 
side  would  make  a  centimetre.  But  this  is  entirely  contrary 
to  the  mathematician's  idea  of  a  point. 

If  a  line  is  composed  of  points,  the  number  of  points 
certainly  cannot  be  finite;  otherwise,  if  the  number  happened 
to  be  odd,  the  line  could  not  be  bisected.  Again,  if  the  side 
and  the  diagonal  of  a  square  each  contained  a  finite  number  of 
points,  they  would  bear  a  definite  numerical  ratio  to  each 
other,  and  this  we  know  they  do  not.  The  existence  of  in- 
commensurables  proves,  in  fact,  that  every  finite  line  must, 
if  it  consists  of  points,  contain  an  infinite  number.  In  other 
words,  if  we  were  to  take  away  the  points  one  by  one, 
we  should  never  take  away  all  the  points,  however  long  we 
continued  the  process.  The  number  of  points  therefore 
cannot  be  counted.  This  is  the  most  characteristic  property 
of  the  infinite  collection — that  it  cannot  be  counted. 

Consider  two  concentric  circles.  From  any  number  of 
points  on  the  circumference  of  the  outer  circle,  draw  radii 
to  the  common  centre.  Each  radius  cuts  the  circumference 
of  the  inner  circle,  so  that  there  is  a  one-to-one  correspondence 
between  all  the  points  on  the  outer  circle  and  all  the  points 


422  CRAFTSMANSHIP  IN  MATHEMATICS 

on  the  inner.  Imagine  the  outer  circle  to  be  so  large  as  to 
extend  to  the  stars,  and  the  inner  one  to  be  so  small  as  to 
be  only  just  visible  to  the  naked  eye.  Further,  imagine 
an  indefinitely  large  number  of  points  packed  closely  round 
the  circumference  of  the  big  circle,  and  all  the  radii  drawn; 
the  number  of  corresponding  points  on  the  inner  circum- 
ference must  be  the  same  as  the  number  on  the  outer. 
Clearly,  in  any  line  however  short,  there  are  more  points 
than  any  assignable  number.  However  large  a  number  of 
points  we  imagine  in  a  line,  no  one  of  them  can  be  said 
to  have  a  definite  successor,  for  between  any  two  points, 
however  close,  there  must  always  be  others. 

Again,  consider  the  class  of  positive  integers.  They  may 
be  put  into  one-to-one  correspondence  with  the  class  of 
all  even  positive  integers,  by  writing  the  classes  as  follows: 

1|2|3|4|5|6| 

2     I     4     I     6     I     8     I   10     I   12     I 

To  any  integer  a  of  the  first  class  there  corresponds  an  integer 
20  of  the  second.  Hence  the  number  of  all  finite  numbers 
is  not  greater  than  the  number  of  all  even  finite  numbers. 
Evidently  we  have  a  case  of  the  whole  being  not  greater 
than  its  part. 

Thus  we  have  another  characteristic  of  classes  called  in- 
finite: a  class  is  said  to  be  infinite  if  it  contains  apart  which  can 
be  put  into  a  one-to-one  correspondence  with  the  whole  of  itself. 

It  is  possible  to  imagine  any  number  of  sequences  whose 
numbers  have  a  one-to-one  correspondence  with  all  the 
integers,  for  instance  all  the  multiples  of  3,  or  of  7,  or  of  97. 
The  characteristics  of  all  such  sequences  are:  (1)  there  is 
a  definite  first  number;  (2)  there  is  no  last  number;  (3)  every 
number  has  a  definite  successor.  Hence  they  must  all  be 
supposed  to  have  the  same  infinite  number  of  members. 

It  is  important  to  notice  that,  given  any  infinite  collection 
of  things,  any  finite  number  of  things  can  be  added  or  taken 
away  without  increasing  or  decreasing  the  number  in  the 
collection. 


THE  CALCULUS:    SOME  FUNDAMENTALS       423 

It  will  be  agreed  that  the  nature  of  an  infinite  number  is 
beyond  the  conception  of  an  ordinary  boy,  and  the  boy 
should  not  be  allowed  to  use  the  term.  Even  the  ordinary 
teacher  may  ponder  over  the  paradox  of  Tristram  Shandy. 
— A  man  undertakes  to  write  a  history  of  the  world,  and  it 
takes  him  a  year  to  write  up  the  events  of  a  day.  Obviously 
if  he  lives  but  a  finite  number  of  years,  the  older  he  gets 
the  further  away  he  will  be  from  finishing  his  task.  If,  how- 
ever, he  lives  for  ever,  no  part  of  the  history  will  remain 
unwritten.  For  the  series  of  days  and  years  has  no  last  term; 
the  events  of  the  nth  day  are  written  in  the  nth  year.  Since 
any  assigned  day  is  the  nth,  any  assigned  day  may  be  written 
about,  and  therefore  no  part  of  the  history  will  remain  un- 
written. 

Neither  Tristram  Shandy  nor  Zeno  is  meat  for  babes,  but 
there  are  certain  elementary  considerations  of  number  se- 
quences with  which  boys  should  be  familiar. 

"  Number  "  in  the  more  general  sense  means  simply 
the  ordinary  integers  and  fractions  of  arithmetic.  All  numbers 
in  mathematics  are  based  on  the  primitive  series  of  integers. 
A  fraction  is,  strictly  speaking,  a  pair  of  integers,  associated 
in  accordance  with  a  definite  law.  This  law  enables  us  to 
substitute  for  each  single  integer  a  pair  of  integers  which 

are  to  be  taken  as  equivalent  to  it.   Thus  -  is  equivalent  to  5. 

In  this  way  we  get  an  infinite  number  of  numerical  rationals 
of  the  same  form. 

Between  any  two  numbers  of  the  sequence  of  integers, 
there  is  an  infinite  number  of  rationals.  For  instance,  between 
8  and  13  there  is  an  infinite  series  of  rationals,  or  between  8  and 
9.  Obviously,  then,  the  rationals  between,  say,  8  and  9  form 
a  sequence  that  is  endless  both  ways.  Between  8  and  9  we 
have,  for  instance,  8-5;  between  8  and  8-5  we  have  8-25;  between 
8-25  and  8-5  we  have  8-375;  and  so  on  indefinitely.  Con- 
secutive fractions,  that  is,  fractions  between  which,  for  example, 
a  mean  cannot  be  inserted,  are  inconceivable,  just  as  are 
consecutive  points  in  space.  The  integers  8  and  9  are  the 


424  CRAFTSMANSHIP  IN  MATHEMATICS 

first  numbers  met  with  beyond  the  sequence.  We  call  these 
numbers  8  and  9  the  upper  and  lower  limits  of  the  sequence. 
There  is  no  last  rational  less  than  9  and  no  first  rational 
greater  than  8.  It  is  erroneous  to  say  that  the  terms  of  a 
sequence  ultimately  coincide  with  the  limit.  The  limit  is 
always  outside  the  sequence  of  which  it  is  the  limit. 

Consider  the  sequence  2  —  y,  2  —  J,  2  —  J,  .  .  .  2  —  -, 

n 

as  n  increases  endlessly.  Here  successively  higher  integral 
values  form  a  sequence  of  rationals  which  constantly  rise  in 
value  but  have  no  last  term.  There  is,  however,  a  rational 
number,  2,  which  comes  next  after  all  possible  terms  of  the 
sequence.  That  is  to  say,  if  any  rational  number  be  named 

less  than  2,  there  will  always  be  some  term  less  than  2  —  - 

n 

between  it  and  the  number  2.  This  is  what  is  meant  by 
calling  2  the  limit  of  the  sequence. 

A  and  B  are  a  given  distance  apart,  say  2".  We  attempt 
to  reach  B  from  A  by  taking  a  series  of  steps,  the  first  step 
being  half  the  whole  distance;  the  second,  half  the  remaining 
distance;  the  third,  half  the  still  remaining  distance;  and 
so  on.  When  would  B  be  reached?  Obviously  the  answer  is 
never.  For  any  step  taken  is  only  half  the  distance  still 
remaining.  Thus  the  successive  distances,  in  inches,  are, 
1,  £,  J,  J,  y£,  and  so  on.  However  many  of  these  distances 
are  added  together,  the  sum  would  never  be  equal  to  the 
whole  distance,  2".  It  is  thus  absurd  to  talk  about  summing 
a  series  to  infinity.  The  limit,  2,  is  not  a  member  of  the 
series;  it  is  unreachable  and  stands,  a  challenger,  right  out- 
side the  series  ,  as  a  limit  always  does. 

The  example  is  well  worth  pursuing. 

The  sum  of  the  series  1  +  1  +  I  +  .  .  .  +  A  =  2  -  Jj. 


The  sum  of  the  series  l  +     +       +...+-  =  2-. 


THE  CALCULUS:    SOME  FUNDAMENTALS      425 

We  can,  of  course,  take  a  number  of  terms  of  the  series  that 
will  be  great  enough  to  make  the  sum  fall  short  of  2  by  less 
than  any  fraction  that  can  be  named,  say  less  than  1/100,000. 

We  have  to  calculate  the  value  of  n  so  that  —  may  be  equal 

to  or  less  than  1/100,000.    By  trial  we  find  —  =  and 

11  216       65536 


2"  =  131072*  The  lattef  is  leSS  than  1/100'000-  Thus  if 
we  take  17  terms  of  the  series,  the  sum  differs  from  2  by 
less  than  1/100,000. 

It  is  impossible  to  take  enough  terms  to  make  the  sum 
equal  to  2.  There  is  always  a  gap  l/2n.  However  great  n  may 
be,  l/2n  is  always  something;  it  is  never  zero.  We  may  say 
that  the  sum  of  n  terms  becomes  nearer  and  nearer  2  as  n 
becomes  greater  and  greater,  or  that  it  tends  towards  2  as  n 
becomes  indefinitely  great.  Do  not  use  the  term  infinity. 

Consider  another  example,  the  decimal  •11111.... 

•11111...  =  ye  +  Too    i"  Tooo"  ~r  Too  (To  ~i~  ^c' 

The  sum  of  the  first  2  terms  = =  — i-. 

100  9-^- 

The  sum  of  the  first  3  terms  = 


The  sum  of  the  first  4  terms  — 


1000         9T|T* 
1111  _       1 
10000  ~  9-' 


By  increasing  the  number  of  terms,  the  sum  can  be  made  to 
differ  by  less  and  less  from  |,  and  this  difference  can  be  made 
smaller  than  any  quantity  that  can  be  named.  Hence  ^  is 
the  limit  to  which  the  sum  tends,  though  this  limit  can  never 
be  actually  reached. 

So  in  cases  like  the  area  under  a  curve.  Where  we  say  that 
PM  is  the  limit  of  the  ordinate  pm,  we  mean  that  by  taking 
mM  constantly  smaller,  pm  may  be  brought  constantly 
nearer  PM,  and  that  it  never  occupies  a  position  so  near  that 


426  CRAFTSMANSHIP  IN  MATHEMATICS 

it  could  not  be  still  nearer.    It  always  remains  the  opposite 

side  of  the  rectangle,  and  never  actually  coincides  with  PM, 

for  PM  stands  outside  all  pos- 
sible positions  of  pm.  But  PM 
is  the  first  ordinate  that  stands 
outside  the  series,  that  is,  there 
is  no  other  ordinate  between 
PM  and  the  series  of  all  possible 
positions  of  pm. 
Fig  24S  The  teacher  should  devise 

other  illustrations  of  the  nature 

of  a  limit.    The  notion  is  fundamental,  and  the  pupils  must 

understand  it. 

Rate 

The  old  geometers  were  concerned  more  with  drawing 
tangents  to  curves,  and  with  finding  the  areas  enclosed  by 
curves,  than  with  rate  of  change  in  natural  phenomena; 
but  the  latter  idea  as  well  as  the  former  one  was  certainly 
in  Newton's  mind,  and  was  embodied  in  the  language  of  the 
calculus,  as  we  now  call  it,  which  he  and  Leibniz  invented. 
The  two  ideas,  tangency  and  rate,  are  virtually  just  two  facets 
of  the  same  idea,  and  in  teaching  the  calculus  the  two  should 
be  kept  side  by  side. 

Pupils  will  have  learnt  something  already  about  tangency. 
And  if  they  have  begun  dynamics,  as  they  ought  to  have  done, 
they  will  have  some  idea  of  the  nature  of  "  Rate  ".  Rate  is 
one  of  those  rather  subtle  terms  which  are  much  better  con- 
sistently used  than  formally  defined.  Even  in  the  lower  Forms, 
boys  should  be  given  little  sums  in  which  the  term  is  correctly 
used:  "  at  what  rate  was  the  car  running?"  and  so  forth. 
But  before  the  calculus  is  begun,  the  notion  of  rate  must  be 
clarified.  This  means  presenting  the  notion,  in  some  way, 
in  the  concrete.  Practical  work  is  essential,  even  if  the  experi- 
ments are  only  of  a  rough  and  ready  character.  Suitable 
experiments  are  described  in  any  modern  book  on  dynamics. 

Consider  a  train  in  motion.    How  can  we  determine  its 


THE  CALCULUS:    SOME  FUNDAMENTALS       427 


velocity  at  some  instant,  say  at  noon?  We  might  take  an 
interval  of  5  minutes  which  includes  noon,  and  measure  how 
far  the  train  has  gone  in  that  period.  Suppose  we  find  the 
distance  to  be  5  miles;  we  may  then  conclude  that  the  train  was 
running  at  60  miles  an  hour.  But  5  miles  is  a  long  distance, 
and  we  cannot  be  sure  that  exactly  at  noon  the  train  was 
running  at  that  speed.  At  noon  it  may  have  been  running 
70  miles  an  hour,  or  perhaps  50  miles,  going  downhill  or 
uphill  at  that  time.  It  will  be  safer  to  work  with  a  smaller 
interval,  say  1  minute,  which  includes  noon  (perhaps 
half  a  minute  before  to  half  a  minute  after  Big  Ben  begins 
to  strike),  and  to  measure  the  distance  traversed  during 
that  period.  But  even  greater  accuracy  may  be  required: 
one  minute  is  a  rather  long  time.  In  practice,  however,  the 
inevitable  inaccuracy  of  our  measurements  makes  it  useless 
to  take  too  small  a  period,  though  in  theory  the  smaller 
the  period  the  better,  and  we  are  tempted  to  say  that  for  ideal 
accuracy  an  "  infinitely  small  "  period  is  required.  The  older 
mathematicians,  Leibniz  in  particular,  yielded  to  this  tempta- 
tion, and  so  gave  wrong  explanations  of  the  working  of  the 
new  mathematical  instrument  (the  calculus)  which  they 
invented. 

Revise  rapidly  some  of  the  easier 
graph  work  and  show  how  change  of 
rate  is  indicated  by  change  of  steep- 
ness in  the  slope. 

The  careful  study  of  a  falling  body 
will  go  far  to  make  clear  the  notion  of 
rate.  Refer  to  Galileo's  experiments 
on  falling  bodies.  Generally  speaking, 
it  will  not  be  possible  to  repeat  such 
experiments,  and  so  obtain  first-hand 
data;  the  necessary  data  must  therefore 
be  provided  otherwise. — Let  fig.  246#  represent  the  path  of 
a  body  falling  from  a  tower  or  down  a  well.  The  three 
lines  allow  the  three  sets  of  values  (distances,  velocities,  times) 
to  be  shown  in  parallel,  the  distances  and  velocities  being 


DISTANCES    VELOCITIES       TIMES 
(FT)  (Frpers«c.)      fseca) 


.64 64 


256 1(28 J 

Fig.  2460 


428 


CRAFTSMANSHIP  IN  MATHEMATICS 


DISTANCES    VELOCITIES     TIME 
(Fr)  (FT  parse^   (sec,) 


shown  at  the  end  of  1,  2,  3,  and  4  seconds,  respectively.  Use 
the  data  to  verify  (perhaps  in  some  degree  to  establish)  the 
formulae  v  =  ft ,  v  =  M  +  /*>  *  =  i(#  +  *0*>  5  =  £/*2« 

Boys  are  often  puzzled  about  the  32  (the  acceleration 
constant).  In  the  first  place,  it  is  a  power  of  2,  and  they  confuse 
it  with  t2.  (It  is  really  best  to  use  the  nearer  value  32-2, 
even  though  the  arithmetic  is  a  little  more  difficult.)  In  the 
second  place  the  boys  are  apt  to  forget  that  this  acceleration 
number  is  merely  the  value  attached  to  a  particular  interval 
of  time,  viz.  1  second.  They  should  be  given  a  little  practice 
with  smaller  intervals,  say  J  seconds. 
The  second  figure,  a  modification  of  the 
previous  figure,  is  therefore  useful.  It 
represents  the  happenings  in  the  first 
second,  at  quarter-second  intervals. 
Since  the  same  amount  of  extra  velocity 
is  added  on  per  second,  we  have  to  take 
one-quarter  of  this  for  each  quarter  of 
a  second.  Observe  that  although  this 
figure  really  represents  the  happenings 
in  the  first  second  of  the  previous  figure, 
the  two  figures  have  an  identical  appear- 
ance so  far  as  the  line-divisions  are  con- 
cerned. The  one  second  is  divided  up  exactly  as  the  four 
seconds  were  divided  up.  It  impresses  boys  greatly  that  this 
sort  of  magnification  or  photographic  enlargement  might  go 
on  "  for  ever  ".  If,  for  instance,  we  take  the  first  quarter- 
second  of  the  last  figure  (2466),  and  magnify  the  distance 
line  16  times  (as  we  did  in  the  case  of  fig.  246a),  we  get  still 
another  replica,  this  time  with  the  quarter-second  divided 
up  to  show  the  happenings  during  each  sixteenth-second. 
However  short  the  distance,  there  is  acceleration,  and  the 
acceleration  has  a  constant  value.  The  acceleration  is 
"  uniform  ". 

"  Uniform  acceleration  is  measured  by  the  amount  by 
which  the  velocity  increases  in  unit  time." — Many  boys 
have  difficulty  in  understanding  what  "  uniform  "  accelera- 


Fig.  2466 


THE  CALCULUS:    SOME  FUNDAMENTALS       429 


Fig.  247 


tion,  such  as  acceleration  due  to  gravity,  really  implies. 
"  If  only  you  would  accelerate  by  adding  on  velocity  in 
definite  chunks  at  equal  intervals,  we  could  understand  it." 

Let  the  boy  have  his  definite  chunks,  at  first,  and  utilize 
these  for  approaching  the  main  idea.  Go  back  to  the  graph. 

Suppose  a  train  to  move  for  1  minute  at  a  uniform  velocity 
of  5  miles  an  hour;  then  to  be  suddenly  accelerated  to  10 
miles  an  hour  and  to  travel  for  1  minute 
at  that  velocity;  then  to  be  accelerated  to  50 
15  miles  an  hour  for  a  third  minute;  to  20 
miles  an  hour  for  a  fourth  minute;  to  25  for 
a  fifth;  and  to  30  for  a  sixth.  How  far  would 
it  have  travelled  altogether? — A  velocity- 
time  graph  shows  this  at  once.  The 
number  of  units  of  area  under  the  graph  is 
1  +  2  +  3  +  4  +  5  +  6-21,  and  this 
gives  us  the  number  of  miles  travelled. 

The  dotted  line  passing  through  the  top  left-hand  corners 
of  the  rectangles  can  easily  be  proved  to  be  straight,  and 
this  evidently  indicates  some  sort  of  uniformity  in  the  motion. 
But  the  whole  of  the  area  under  this  line  is  not  enclosed  by 
the  rectangles;  there  are  6  little  triangles  unaccounted  for. 
How  are  these  triangles  to  be  explained?  By  the  fact  that 
really  we  have  imagined  an  impossible  thing,  viz.  that  at 
certain  times  the  train's  speed  was  instantaneously  increased 
5  miles  an  hour. 

Now  although  in  practice  we  know  that  even  in  the  very 
best  trains  acceleration  is  really  brought  about  by  sudden 
jerks,  these  jerks  are  virtually  imperceptible,  and  it  is  there- 
fore not  impossible  to  imagine  an  acceleration  free  from  such 
sudden  increases.  It  may  be  easily  illustrated  by  running 
water:  the  following  ingenious  illustration  we  owe  to  Professor 
Nunn. 

Attached  to  my  bath  is  a  tap  so  beautifully  made  that 
by  means  of  the  graduated  screw-head  I  can  regulate  the 
amount  of  water  running  in  up  to  8  gallons  a  minute. 


430 


CRAFTSMANSHIP  IN  MATHEMATICS 


I  turn  on  the  tap  for  one  minute,  the  water  running  at 
the  rate  of  1  gallon  a  minute;  in  that  time  1  gallon  has  been 
delivered.  Then  I  turn  the  tap  on  further,  to  deliver  water 
at  the  rate  of  2  gallons  a  minute,  and  allow  it  to  run  for  one 
minute;  during  this  minute,  2  gallons  have  been  delivered. 
Thus  I  continue  for  8  minutes,  8  gallons  running  in  during 
the  eighth  minute.  The  graph  (fig.  248,  i)  shows  the  water 
run  in  during  the  successive  minutes;  the  shaded  rectangle, 
for  instance,  represents  the  amount  of  water  (5  gallons) 


GALLS  s 

per 
M.N. 


MINUTES 


23456 

MINUTES 


Fig.  248 


run  in  during  the  fifth  minute.     Total  number  of  gallons 
delivered  =  36. 

I  now  repeat  the  operation,  but  this  time  I  turn  the  tap 
on  every  half  minute,  beginning  by  running  in  \  gall,  a 
minute,  and  increasing  by  \  gall,  each  half  minute.  The 
first  delivery  will  be  J  gall.,  the  next  \  gall.,  and  so  on,  the 
last  being  4  gall.  But  note  (fig.  248,  ii)  that  during  the  last 
half  of  the  fifth  minute,  when  2|  gall,  were  delivered,  the  rate 
of  delivery  was  5  gall,  a  minute;  this  column  has  the  same 
height  as  the  corresponding  column  in  (i),  but,  of  course, 
only  half  its  area.  The  rate  of  flow  during  the  half  minute 
was  the  same,  though  only  half  the  5  gall,  was  actually 
delivered.  The  rate  of  flow  during  the  last  half  of  the  eighth 
minute  was  8  gall,  a  minute,  though  only  4  gall,  were  de- 
livered. Total  number  of  gallons  delivered  =  34. 


THE  CALCULUS:    SOME  FUNDAMENTALS       431 


I  repeat  again,  this  time  allowing  the  water  to  be  in- 
creased every  J  minute,  beginning  by  running  in  J  gall, 
a  minute,  and  increasing  by  J  gall,  each  J  minute.  The 
first  delivery  is  thus  yg-  gall.,  the  next  J  gall.,  the  last  f|-  or 
2  gall.  Note  (fig.  iii)  that  during  the  last  |  of  the  fifth  minute, 
when  1J  gall,  were  delivered,  the  rate  of  delivery  was  still 
5  gall,  a  minute;  the  column  has  the  same  height  as  the 
corresponding  columns  in  the  first  two  figures,  but  of  course 
only  J  of  the  area  of  the  column  in  the  first  figure.  The 

8 
7 

G 


O    2 


I      Z 


3456 

MINUTES 
(in) 


1234 

MINUTES 
(iv) 


676 


Fig.  249 


rate  of  flow  in  the  column  preceding  HK  was  the  same  in  all 
3  cases.  Total  number  of  gallons  delivered  (fig.  249,  iii)  =  33. 

I  repeat  the  operation  once  more,  this  time  turning  on  the 
tap  gradually  and  continuously,  in  such  a  way  that  at  the 
end  of  the  first  minute  the  water  is  running  at  the  rate  of 
1  gall,  a  minute,  though  only  momentarily;  and  so  on. 
At  the  end  of  the  eighth  minute  I  turn  off,  i.e.  at  the  very 
moment  when  the  rate  of  flow  has  reached  8  gallons  a  minute. 
The  graph  (fig.  249,  iv)  is  now  a  straight  line,  and  its  area  is 
|(8  X  8)  or  32  units,  the  number  of  gallons  delivered. 

Observe  that,  in  all  4  figures,  the  rate  of  delivery  at  the 
end  of  any  particular  minute  is  the  same,  for  instance  at 
the  end  of  the  fifth  minute,  represented  by  HK;  though  in 


432  CRAFTSMANSHIP  IN  MATHEMATICS 

the  last  case,  when  the  tap  is  gradually  turned  on,  the  rate 
at  any  particular  time  is  only  momentary,  since  the  rate  is 
continuously  changing. 

In  the  last  figure,  HK  no  longer  bounds  a  rectangle,  as 
it  did  in  the  previous  three  figures;  all  the  columns  have 
become  indefinitely  narrow.  The  column  which  HK  bounded 
has  shrunk  to  a  mere  line  which  therefore  cannot  represent 
any  quantity  of  water  delivered.  Still,  as  it  has  the  same 
height  as  the  series  of  gradually  narrowing  columns  which 
it  bounded,  we  say  that  it  represents  a  rate  of  flow  of  5  gall. 
a  minute,  just  as  the  columns  did.  But  this  rate  of  flow 
is  clearly  not  a  rate  of  flow  during  any  interval  of  time, 
however  small.  Hence  we  say  it  is  the  rate  of  flow  at  the 
end  of  the  fifth  minute. 

Boys  ought  now  to  understand  clearly  that  the  velocity 
of  a  body  at  any  instant  is  measured  by  the  rate  per  unit 
time  in  which  distance  is  being  traversed  by  the  body  when 
in  the  immediate  neighbourhood  of  that  instant. 

A  body  cannot  move  over  any  distance  in  no  time,  so  that 
we  could  not  find  its  velocity  by  observing  its  position  at 
one  single  instant.  To  find  its  rate  of  motion,  we  must 
observe  the  distance  traversed  during  some  interval  of  time 
near  the  given  instant,  this  interval  of  time  being  the  shortest 
possible.  Hence  the  term  velocity  at  any  instant  must  be 
regarded  as  an  abbreviation  for  average  velocity  during  a 
very  small  interval  of  time,  including  the  given  instant.  But 
we  have  no  means  of  finding  such  a  velocity  by  actual  experi- 
ment. We  have  to  adopt  other  means. 

It  is  sometimes  said  that  acceleration  at  a  given  instant 
of  time  is  measured  by  the  rate  per  unit  time  at  which  the 
velocity  is  increasing  in  the  immediate  neighbourhood  of  the 
given  instant,  or  the  average  acceleration  in  a  small  interval 
of  time  including  the  given  instant. 

The  question,  what  is  meant  by  the  statement  that  at  a 
certain  moment  a  thing  is  moving  at  the  rate  of  so  many 
feet,  a  second  ought  now  to  be  answered  by  all  average 


THE  CALCULUS:    SOME  FUNDAMENTALS       433 


pupils.  Sixth  Form  boys  should  grasp  the  full  significance 
of  the  following  formal  statement:  "  if  the  magnitude 
possessed  by  any  increasing  or  decreasing  quantity  be  re- 
presented by  an  area-function,  the  rate  of  increase  or  decrease 
of  the  quantity  at  any  specified  point  is  given  by  the  corre- 
sponding ordinate  function/' 

Thus  if  any  given  function  is  regarded  as  an  area-function, 
the  corresponding  ordinate  function  may  be  called  the 
rate-function  of  the  former. 

Calculation  of  Rate -functions 

We  may  consider  again  the  rate-function  corresponding 
to  the  area-function  ax3.  According  to  the  results  at  the 
end  of  the  last  chapter,  this  should  be  3ax2. 


Q X 


(i) 


Fig.  250 


Let  the  curve  in  fig.  250  have  the  property  that  the  area 
under  it  from  the  y  axis  up  to  any  ordinate  PQ  is  ax3.  How 
may  we  determine  the  exact  height  of  PQ? 

Take  two  other  ordinates  (fig.  ii)  CD,  EF,  each  at  distance 
h  from  PQ.  Draw  upon  DQ,  QF  rectangles  whose  areas  are 
respectively  equal  to  those  of  the  strips  under  the  curve 
between  CD  and  PQ,  and  PQ  and  EF.  Let  the  curve  cut 
the  upper  ends  of  these  rectangles  in  p  and  p'.  Draw  the 
ordinates  pq  and  p'q'. 

(E201  29 


434 


CRAFTSMANSHIP  IN  MATHEMATICS 


Although  we  cannot  calculate  PQ  directly,  it  is  easy  to 
calculate  pq  and  p'q1 '.    We  have: 


pq  X  h  =  area  CQ. 
.  _  area  CQ 
•mpq  h~ 

_  ax9  —  a(x  —  / 

h 

=  (3*2  -  3xh  + 
i.e.  /><?  -  {3*2  -  h(3x  - 


X  h  =  area  EQ. 

..//__  area  EQ 
.£<,  _ 

a(x  +  /O3 


i.e.  p'q'  =  {3*2  +  £(3*  +  /*)}«• 


Whatever  value  ^  may  have,  h  may  be  taken  smaller;  hence 
h  must  be  smaller  than  3x.  Thus  h(3x  —  A)  and  A(3#  +  h) 
must  both  be  positive,  and  pq  will  necessarily  be  less  than 
PQ,  and  p'q'  greater  than  PQ.  By  making  h  small  enough, 
we  can  make  pq  and  p'q'  differ  from  Sax2  as  little  as  we  please. 
In  other  words,  PQ  must  lie  between  all  possible  positions 
of  pq  and  p'q' y  and  thus  the  value  3ax2  is  the  only  value  left 
for  it  to  possess. 

The  Rate  as  a  Slope. — Here  is  another  way  of  consider- 
ing a  rate-function.  Let  OQ'  be  the  curve  y  =  ax3.  Let 
the  abscissa  of  any  point  P  be  x,  and  the  abscissae  of  two 
neighbouring  points  Q  and  Q',  x  —  h  and  x  +  h,  respectively. 
While  x  increases  from  x  -—  h  to  xy  and  from  x  to  x  +  A, 
y  increases  by  Q<?  and  q'Q',  respectively.  (Fig.  251.) 

Hence  the  average  rate  of  the  latter  increases  must  be 

S*?  and  ?yi,  i.e.  tanP*X  and  tanPz'X,  respectively. 


___  PM  -  QN 

~~         h 

_  ax*  -  a(x  -  Kf 

h 
i.e.  tan  P*X  =  {3*2  -  h(3x  -  h)}a. 


tan 


h 
Q'N'  -  PM 


-  ax3 


i.e.  tan 


>?  +  h)}a, 


(both  as  in  the  last  example) 
As  h  gets  smaller,  Q  and  Q'  approach  P,  tanP/X  being  always 


THE  CALCULUS:    SOME  FUNDAMENTALS       435 

less,  and  tanP^'X  always  greater,  than  Sax2,  though  by  taking 
h  small  enough,  they  may  be  made  to  differ  as  little  as  we 
please  from  Sax2. 

If  PT  be  drawn,  so  that  tanPTX  =  3a*2  exactly,  then 
PT  is  evidently  the  tangent  at  P.  For  a  line  through  P  ever 
so  little  divergent  from  PT  would  make  with  the  x  axis  an 
angle  greater  or  less  than  PTX,  and  so  would  cut  the  curve 


h,  is  purposely  exagge rated 
ib  make  the  -Kdure  clear 


Fig.  251 

in  one  of  the  possible  positions  of  Q  or  of  Q'.  Hence  PT  is 
the  only  line  which  meets  the  curve  at  P  but  does  not  cut  it. 

Thus  PT  holds  among  secants  such  as  PQ  or  PQ'  the 
same  unique  position  that  HK  holds  among  the  rectangles 
(figs.  248,  249),  or  that  PQ  holds  amongst  the  other  ordinates 
(fig.  250). 

In  fig.  251,  the  slopes  of  PQ  and  PQ'  measure  the  average 
rate  of  change  of  the  function  during  the  changes  of  x  repre- 
sented by  NM  and  N'M.  The  slope  of  PT  does  not  measure 
the  change  during  any  intervals,  but  evidently  measures  what 
has  been  defined  as  the  rate  of  change  of  the  function  at  the 
moment  (or  for  the  value  of  x)  represented  by  OM. 


436  CRAFTSMANSHIP  IN  MATHEMATICS 

Meaning  of  "  Limit  " 

The  common  element  in  the  three  cases  considered, 
HK  (figs.  248,  249),  PQ  (fig.  250),  PT  (fig.  251),  is  described 
by  saying  that  all  three  are  examples  of  a  limit.  In  all 
three  cases,  members  of  a  series  have  been  brought  nearer 
and  nearer  the  limit,  but  they  have  never  been  so  near  that 
they  could  not  have  been  brought  nearer.  They  have  always 
remained  "  in  the  neighbourhood  "  of  the  limit,  but  in 
every  case  the  limit  has  been  unreachable.  In  all  three  cases, 
the  limit  is  the  first  number  outside  the  series. 

A  rate-function  is  sometimes  given  this  general  definition: 
Take  the  given  function  of  x,  and  find  how  much  its  value 
changes  when  x  is  raised  or  lowered  by  any  positive  number 
h.  Divide  this  change  by  h,  and  so  obtain  the  average  rate 
of  change  for  a  change  of  the  variable  from  x  —  h  to  hy  or 
from  x  to  x  +  h.  The  rate-function  is  the  limit  of  the  quotient 
and  is  indicated  more  and  more  closely  as  h  gets  smaller 
and  smaller. 

The  Two  Main  Uses  of  Limits 

1.  To  define  the  velocity  of  a  given  point  at  a  given  moment. 
— If  we  define  velocity  as  the  quotient  of  a  distance  travelled, 
by  the  time  in  which  it  is  traversed,  then  "  the  velocity  at 
a  given  moment  "  is  not  a  velocity  at  all. 

On  the  other  hand,  if  we  consider  the  distance  travelled 
by  the  point  during  a  series  of  constantly  decreasing  in- 
tervals of  time,  and  divide  each  distance  by  the  length* of  the 
corresponding  interval,  we  shall  again  fail,  as  a  rule,  to 
obtain  anything  that  can  be  called  the  velocity  of  the  point, 
for^all  the  results  will  be  different,  except  in  the  special  case 
of  uniform  motion.  But  if  the  sequence  of  average  velocities 
thus  calculated  follows  some  definite  law  of  succession  as 
the  interval  is  taken  smaller,  then  it  will  generally  have  a 
definite  limit  as  the  interval  approaches  zero.  Thus  the 
limit  is  a  perfectly  definite  number,  associated  in  a  perfectly 


THE  CALCULUS:    SOME  FUNDAMENTALS       437 

unambiguous  way,  both  with  the  given  moment  and  with 
the  endless  sequence  of  different  average  velocities.  More- 
over, for  small  intervals  of  time,  the  average  velocities  are 
sensibly  equal  to  the  limit,  the  differences  being  of  theoretical 
rather  than  of  practical  importance.  It  follows  that  although 
the  "  velocity  at  the  given  moment  "  is  not  really  a  velocity 
at  all,  it  is  quite  the  most  useful  number  to  quote  in  order 
to  describe  the  behaviour  of  the  moving  point  while  it  is 
in  the  neighbourhood  of  the  place  which  it  occupies  at  the 
given  moment. 

2.  To  determine  a  magnitude  which  cannot  be  evaluated 
directly. — Consider  again  fig.  250.  We  had  to  determine  the 
height  of  the  ordinate  PQ.  We  found  (i)  that  it  lies  between, 
and  is  the  limit  of,  a  lower  sequence  consisting  of  ordinates 
pq  and  an  upper  sequence  consisting  of  ordinates  p'q'\  (ii) 
that  it  lies  similarly  between,  and  is  the  limit  of,  the  se- 
quences of  numbers  represented  by  {  3#2  —  h(3x  — -  h) }  a 
and  {  3#2  -f-  h(3x  +  h)  }  a\  and  (iii),  that  the  latter  sequence 
corresponds  to  the  former,  term  by  term.  From  these 
premises  it  seems  to  be  an  inescapable  conclusion  that  the 
height  of  PQ  is  exactly  3##2,  for  PQ  is  the  only  line  between 
the  two  sequences  of  (i),  and  3ax2  is  the  only  number  between 
the  two  sequences  of  (ii). 

For  blackboard  revision  work  occasionally,  devise  questions 
to  emphasize  these  principles  (the  term  gradient  might  now 
be  used  generally): 

(1)  The  gradient  of  a  chord  is  the  average  gradient  of  the 
arc. 

(2)  A  tangent  is  the  limiting  position  of  a  secant. 

(3)  The  gradient  of  a  tangent  at  P  is  the  gradient  of  the 
curve  at  P. 

(4)  The  gradient  of  the  tangent  is  the  rate  at  which  the 
function  is  changing. 

(5)  The  limiting  value  of  the  slope  of  a  secant  is  the  slope 
of  the  tangent. 

"  In   the  neighbourhood  of." — We   have  spoken   of  the 


438  CRAFTSMANSHIP  IN  MATHEMATICS 

members  of  a  series  being  "  in  the  neighbourhood  of "  a 
limit.  What  is  a  neighbour!  That  is  a  question  of  degree. 
In  Western  Canada,  a  man's  nearest  neighbours  might  be 
40  or  50  miles  away;  in  an  English  country  district,  perhaps 
a  single  mile;  in  a  town,  only  a  few  yards;  round  one's  own 
table,  only  a  few  inches.  So  with  number  sequences:  it  is 
just  a  question  of  degree.  For  instance,  we  know  that  TT  comes 
within  the  interval  3-1  and  3-2,  and  therefore  3-1  and  3-2 
are  neighbours  of  TT.  But  TT  also  comes  within  the  smaller 
interval  3-13  and  3-15,  which  are  therefore  closer  neigh- 
bours of  77.  Again,  TT  comes  within  the  interval  3*1414 
and  3*1416,  which  are  therefore  still  closer  neighbours  of  TT. 
And  so  we  might  go  on.  However  close  our  selected  neigh- 
bours of  77,  we  can  always  find  still  closer  neighbours.  Thus 
77  always  has  neighbours  no  matter  how  small  the  interval 
in  which  he  is  enclosed.  It  is  all  a  question  of  standard  of 
approximation.  The  important  thing,  when  dealing  with 
limits,  is  that  we  must  never  think  of  the  interval  shrinking 
to  nothing.  Think  of  the  interval  as  always  large  enough  for 
standing  room  both  for  77  itself  and  some  neighbours.  The 
neighbours  cannot  be  thought  of  as  disappearing  altogether. 

Secant  to  Tangent  Again 

The  gradient  of  a  straight-line  graph  AB  is  determined 
easily  enough:   it  is  the  ratio,  ordinate  j> /abscissa  x.      The 

ratio  may  be  determined 
from  any  selected  bit  of 
the  line.  Or,  if  we  like, 
we  may  increase  the  line, 
say  to  BC,and  take  the  ratio 
CD  (—  increment  of  y)  to 

— • •    BD  (—  increment  of  x). 

Fig.  2S2  But  if  the  graph  is  a 

curve,  the  gradient  at  any 

specified  point  on  the  curve  is  determined  by  the  tangent 
at  that  point.     A  ruler  held  against  the  edge  of  an  ordinary 


THE  CALCULUS:    SOME  FUNDAMENTALS       439 

dish  is,  practically,  a  tangent  at  a  point  on  the  ellipse.  If, 
then,  we  want  to  determine  the  gradient  of  a  curve,  why 
not  just  draw  the  tangent  and  measure  the  angle  it  makes 
with  the  x  axis? 

With  a  circle  this  would  be  easy  enough:  we  should  draw 
a  radius  to  the  point  and  then  a  line  at  right  angles;  and 
there  are  simple  rules  for  certain  other  curves.  But  merely 
to  hold  a  ruler  against  a  curve,  and  to  draw  a  line,  is  not  to 
draw  a  tangent  that  we  can  accept. — Circulate  amongst  the 
class  copies  of  a  mechanically  drawn  parabola,  tell  the  boys 
to  draw  a  tangent  at  the  point  P,  and  then  to  measure  the 
angle  that  the  tangent  makes  with  the  x  axis.  The  angles 
will  probably  be  all  different.  Clearly  the  method  will  not 
do,  for  the  angles  ought  to  be  the  same  in  all  cases. 

If  we  draw  a  secant  instead  of  a  tangent,  and  find  the 
gradient  of  the  secant,  we  shall  evidently  have  the  average 
gradient  of  the  curve  between  the  two  points  P  and  Q  where 
the  secant  intercepts  the  curve.  Would  that  help? 

Yes,  but  if  the  points  are 
far  apart,  as  P  and  Qx,  the 
slope  of  the  secant,  and 
therefore  the  average  gra- 
dient of  the  curve  between 
the  two  points,  differs  much 
from  the  gradient  of  the 
tangent  PT.  If  we  bring 
the  points  closer  together, 
say  P  and  Q2,  the  gradient 
of  the  secant  is  nearer  the 
gradient  of  the  tangent.  If  p. 

we  bring  Q  down  to  Q3  the 

gradient  of  the  secant  PQX  is  still  nearer  the  gradient  of  the 
tangent.  It  is  this  gradient  of  the  tangent  that  we  have  to 
find  somehow. 


[Some  teachers  prefer  that  numerical  considerations  like 
those  that  follow  should  precede  the  more  general   work 


440 


CRAFTSMANSHIP  -  IN  MATHEMATICS 


concerning  the  graph,  as  outlined  in  the  earlier  part  of  this 
chapter.  I  have  seen  equally  satisfactory  final  results  obtained 
from  both  sequences.] 

Let  us  actually  calculate  the  gradients  of  successive 
secants,  and  see  if  we  can  learn  anything  from  the  results. 
On  a  parabola  we  will  select  a  point  P  where  x  =  1  (and 
.".  y  =  I2  —  1),  and  keep  this  fixed.  We  will  also  place  a 
point  Q  on  the  curve,  at  first  where  x  =  1-5  (and  .'.  y  =  (1*5)2 
=  2-25).  Thus,  since  P  is  (1,  1)  and  Q  is  (1-5,  2-25),  the 
increment  of  x  is  '5,  and  the  increment  of  y  is  1'25.  (The 
piece  of  line  PQ  may  be  looked  upon  as  an  "  increase " 
of  the  line  AP;  hence  the  term  "  increment  "  may  usefully 
be  applied  to  the  corresponding  increases  of  x  and  y.) 

The  gradient  of  the  secant  =  ——  = ==  2*5.     Now 

PV         '5 


1-25 


THE  CALCULUS:    SOME  FUNDAMENTALS       441 

bring  Q  gradually  closer  to  P.  Let  the  next  four  x  values  be 
1-4,  1*3,  1-2,  1-1;  then  the  corresponding  y  values  are  (1*4)2, 
(1*3)2,  (1-2)2,  (1*1)2.  The  gradient  calculations  may  be  sum- 
marized thus  (the  x  and  y  increments  are  often  indicated  by 
h  and  kt  respectively): 


*  =  ON  = 

1-5 

1-4 

1-3 

'  1-2 

1-1 

y  =  x*  =  ON2  =  NQ  = 

2-25 

1-96 

1-69 

1-44 

•44 
•2 

1-21 

*  =  QV- 

h  =  PV  =  MN  = 

1-25 
•5 

•96 
•4 

•69 
•3 

•21 
•1 

2-1 

Gradient  -  J  =  ^  -  = 

h       MN 

2-5 

2-4 

2-3 

2-2 

Note  how  the  value  of  the  gradient  has  diminished  from 
2-5  to  2-1.  We  cannot  write  h  =  0,  or  the  denominator  of 
our  ratio  would  equal  0,  and  the  ratio  would  have  no 
meaning.  But  we  may  continue  to  diminish  the  values  of 
the  x  increment,  and  calculate  the  gradient  as  before.  We 
may  make  the  increments  as  small  as  we  please.  Let  us 
calculate  the  gradient  when  the  successive  values  of  x  for  Q 
are  1-01,  1-001,  1-0001,  1-00001,  so  that  the  x  increments 
are  -01,  -001,  -0001,  -00001.  We  cannot  draw  the  figure, 
for  the  increments  are  much  too  small  to  be  shown. 


*  =  ON 

1-01 

1-001 

1-0001 

1-00001 

y  =  x2  =  ON2  =  NQ  = 

1-0201 

1-002001 

1-00020001 

1-0000200001 

fc  =  QV  = 

h  =  PV  -  MN  = 

•0201 
•01 

•002001 
•001 

•00020001 
•0001 

•0000200001 
•00001 

Gradient  -  \  =  §J  -  = 

h      MN 

2-01 

2-001 

2-0001 

2-00001 

442  CRAFTSMANSHIP  IN  MATHEMATICS 

We  observe  (1)  that  however  small  we  make  h  (the  in- 
crement of  x),  the  value  of  the  gradient  always  exceeds  2; 
(2)  that  the  smaller  we  make  the  increment,  the  smaller  is 
the  excess  of  the  gradient  over  2.  Evidently  we  can  approach 
to  within  any  degree  of  approximation  we  like  to  name;  it 
is  only  a  question  of  making  h  small  enough  to  start  with. 
We  observe  also  that  the  more  nearly  the  value  of  the  gradient 
approaches  2,  the  more  nearly  does  the  secant  approach 
the  position  of  the  tangent.  As  long  as  the  secant  remains 
a  secant,  it  can  never  be  a  tangent,  and  it  must  always  have 
a  gradient  in  excess  of  2.  But  the  successive  gradients  seem 
to  compel  us  to  infer  that  the  gradient  of  the  tangent  itself, 
and  therefore  of  the  curve,  is  2.  Thus  we  regard  2  as  the 
limiting  value  of  the  gradient  of  all  possible  secants.  It  is  a 
value  that  is  never  quite  reached  by  any  secant,  for  the  tangent 
stands  alone,  outside  them  all,  four-square  and  defiant! 

Thus  we  have  found  that,  for  the  function  y  =  x2,  the 
gradient  of  the  point  P,  where  x  —  1,  is  2. 

We  may  arrive  at  the  same  result  by  arguing  more  generally, 
merely  calling  the  increments,  h  and  k. 

The  co-ordinates  of  P  are  (1,  1). 

The  co-ordinates  of  Q  are  (1  +  h>  1  +  &)• 

Since  y  =  x2, 

(1  +  *)  =  (1  +  h? 


:.  k  =  2h  +  h2, 

QV      k      2h 
----— 


From    this    point    on,    argument    nowadays    commonly 
proceeds  thus: 

As  Q  approaches  P,  so  h  tends  towards  0.     We  have  to 

k       2/z  -4-  h2 

find  the  limit  to  which  -  or  —  '-—  —  tends  as  h  tends  towards  0 
h  h 

9/j  _1_  /»2 

As  long  as  h  is  4=  0,  Ln  \  n  =  2  +  h,  and  as  h-+  0 


THE  CALCULUS:    SOME  FUNDAMENTALS       443 

2  +  h  -+  2.  If  we  decide  that  2  +  h  must  differ  from  2  by 
less  than  1/1000000,  there  is  no  difficulty;  we  merely  give 
to  h  a  value  less  than  that,  e.g.  1/1000001. 

k 

In  the  limit,  as  h  ->  0,  T  ->  2. 
A 

Hence  the  gradient  of  the  curve  at  P  =  2. 

We  may  now  find  the  gradient  at  any  point  P  (xy  y). 
The  co-ordinates  of  Q  are  (x  +  A,  j>  +  A). 

.'•  (y  +  *)  =  (*  +  A)2 

=  *2  +  2xh  +  A2, 

/.  k  =--  2xh  +  A2. 

Hence  the  gradient  of  PQ  =  ^  =  2x  +  h  if  h  4=  0. 

A 

Now  as  Q  approaches  P,  A  ->  0. 
/.  the  gradient  of  the  curve  at  P  —  limit  of  (2x-\-h)  as  A->  0, 


I  am  not  quite  happy  about  the  language  of  this  argument, 
though  it  is  now  in  common  use  and  has  been  designed  to 
get  over  the  old  difficulty  of  infinitesimals  and  of  the  absurdity 
of  dividing  by  0.  But  even  able  boys  in  the  Sixth  sometimes 
admit  that  the  reasoning  is  not  clear  to  them,  saying  that 
they  feel  they  take  a  leap  over  the  final  gap  to  the  limit. 
The  teacher  must  insist  that  the  gap  is  really  never  crossed, 
that  the  interval  still  remains,  that  the  limit  is  always  there 
with  a  crowd  of  neighbours  who  vainly  strive  to  reach  him; 
that  every  neighbour  has  a  value  a  little  greater  than  2x  (or, 
in  some  of  our  earlier  illustrations,  a  little  less),  and  that  the 
value  2x  is  a  solitary  value,  which  therefore  we  feel  bound  to 
assume  is  the  value  which  belongs  to  the  Limit,  and  to  the 
Limit  alone. 

Revise:  The  function  y  =  x2.  —  To  calculate  the  ordinate 
for  any  value  of  x,  work  out  the  value  of  x2.  To  calculate  the 
gradient  for  any  value  of  x,  work  out  the  value  of  2x. 


444 


CRAFTSMANSHIP  IN  MATHEMATICS 


Thus  x2  may  be  described  as  the  formula  for  the  ordinate, 
and  2x  as  the  formula  for  the  gradient.  In  other  words,  the 
function  x2  gives  the  ordinate,  and  the  function  2x  the  gradient, 
for  any  value  of  x. 

x2  is  the  original  function  which  defines  the  curve;  2x 
is  called  the  derived  function  of  x2.  The  process  of  finding 
the  derived  function  of  a  given  function  is  called  differentiation. 

Since  the  gradient  of  the  tangent  to  y  —  x2  at  any  point 
P  is  2#,  the  gradient  where  x  =  1,  is  2;  where  x  =  2,  is  4; 
where  x  =  3,  is  6;  &c.  Does  this  square  with  the  work  we 
have  done  in  pure  geometry?  We  found  (p.  409), 

j-^r^          ^  r         u  1         i  latus  rectum 

gradient  of  tangent  to  axis  of  parabola  = 
5  *  F 


or  gradient  of  tangent  to  tangent  at  vertex  = 


ordinate 
ordinate 
latus  rectum' 


Let  the  tangent  at  the  vertex  be  the  x  axis,  and  let  the  axis 


of  the  parabola  be  the  y  axis.    Let  S  be  the  focus,  and  let 
the  latus  rectum  LSP  be  unit  length. 


THE  CALCULUS:    SOME  FUNDAMENTALS       445 

Half  the  latus  rectum  =  SP  =  |.  Since  PN  =  half  the 
distance  of  P  from  the  directrix  (not  shown),  PN  =  |PS  =  J. 
Hence  the  co-ordinates  of  P  are  (£,  £). 

At  the  point  Q  (1,  1),  gradient  of  tangent  to  OX 
ordinate          __  1  _  ~ 


half  latus  rectum 

At  the  point  W  (2,  4),  gradient  of  tangent  to  OX 
ordinate         _  2  __  . 


half  latus  rectum       \ 

At  a  point  Z  (3,  9),  gradient  of  tangent  to  OX 

ordinate  3       , 

=  -•  =  o. 


half  latus  rectum       \ 

Clearly  then,  the  new  method  of  finding  the  slope  of  the 
tangent  does  produce  a  result  absolutely  accurate,  not  merely 
approximately  accurate.  Evidently  the  "  limit  "  argument  is 
sound,  though  we  must  always  remember  that  the  limit 
is  outside  the  sequence  under  consideration,  never  reached 
by  any  member  of  the  sequence. 

The  Calculus  Notation 

We  have  used  the  letters  h  and  k  to  denote  the  increases 
("  increments  ")  in  the  values  of  x  and  y.  But  the  increments 
always  actually  considered  are  very  small,  and  the  symbol 
generally  used  to  denote  them  is  the  Greek  letter  delta 
(A  or  8)  prefixed  to  the  value  of  x  or  y  from  which  the  in- 
crement begins.  Pronounce  A*  as  "  delta  x  ";  the  symbol 
A#  must  be  taken  as  a  whole;  A  is  not  a  multiplier  and  has 
no  meaning  apart  from  the  x  and  y  to  which  it  refers. 
Remember,  then,  to  write  Ax  instead  of  /r,  and  by  instead 
of*. 


446  CRAFTSMANSHIP  IN  MATHEMATICS 

A#  means  "  the  increment  of  x  ";    Ay  means  "  the  in- 
crement of  y  ". 

Ay  -          .     increment  of  y     rp,      A,  , 

—  means  the  ratio  -=- .     1  he  A  s  cannot  be 

A#  increment  or  x 

cancelled. 

Treat  — -  exactly  as  if  written  -;   it  measures  the  average 

A#  h 

gradient  of  the  graph   over   the  interval  between   x  and 

x  +  A#. 

The  limit  of  — ^  is  the  gradient  of  the  graph  at  the  point 

.          dy 

given  by  x.     It  is  sometimes   written  D(y),  sometimes   -^-. 

j  ax 

But  the  curious  thing  is  that,  although  ~  looks  like  a  ratio 

dx 

or  a  fraction,  it  is  not  a  ratio  or  fraction.    The  symbols  dy, 
dx,  written  separately,  have  no  meaning.     The  limit  of  A# 

is  not  dx\  the  limit  of  Ay  is  not  dy.   ~-  is  just  a  single  symbol. 

dx 

-~  is  always  a  ratio  of  real  value;    -f-  is  not  a  ratio  at  all 
A#  dx 

and  is  therefore  very  misleading  to  the  eye. 

The  process  of  finding  D(y)  or  —  is  called  differentiation. 
,  ax 

D(y)  or  -f-  has  received  various  names: 
dx 

(1)  The  derivative  of  y  or  f(x)  with  respect  to  x. 

(2)  The  differential  coefficient  of  y  with  respect  to  x. 

(3)  The  derived  function  of  y  with  respect  to  x. 

We  will  differentiate  x*.   Let  y  =  re4.  When  x  is  increased 
to  x  +  A#,  let  jy  be  increased  to  y  +  Ay.  Then: 

y  +  Ay  =  (x  +  A*)4 

=  *4  -f  4*3A*  -f  6*2(A*)a  +  4*(A*)8  +  (A*)4; 
-f  6*2(A*)a  4-  4*(A*)3  -f  (A*)4, 


THE  CALCULUS:    SOME  FUNDAMENTALS       447 

Hence  as  A*  ->  0,  -  ~-  ->  ky?. 

A* 

.'.  ^  =  4*'. 
dx 

Let  the  class  discuss  the  result  (or  one  like  it)  critically.  In 
particular,  discuss  the  significance  of  the  arrows.  Forms 
and  language  that  might  pass  muster  in  an  examination  room 
should  be  subjected  to  the  closest  scrutiny  in  class.  It  is  a 
fact  that  those  boys  who  have  acquired  facility  in  working 
out  the  ordinary  stock  exercises  in  the  calculus  are  often 
nonplussed  when  cross-examined  in  the  underlying  funda- 
mental notions. 

There  can  be  no  doubt  that  the  idea  of  derived  functions 
is  best  introduced  as  a  generalization  of  the  familiar 
ideas  of  connexions  between  area  functions  and  ordinate 
functions,  ordinate  functions  and  gradient  functions,  &c. 

The  notation  -f  should  not  be  introduced  too  soon.     D(y) 
ax  j 

is  much  preferable.    The  symbol  ~-  originated  with  Leibniz 

(not  with  Newton),  and  it  expresses  a  view  of  the  nature  of  a 
differential  coefficient  that  is  out  of  harmony  with  modern 
ideas  and  conflicts  with  the  doctrine  of  limits.  Originally 
the  view  was  that  any  finite  value  of  the  variables  y  and  x 
is  really  the  sum  of  a  vast  number  of  "  infinitesimal  "  values 
which,  though  immeasurably  small,  have  yet  a  definite 

magnitude.     Thus  the  differential  coefficient  -f-  was  looked 

dx 

upon  as  simply  the  ratio  of  the  "  infinitesimals "  of 
two  variables,  the  ratio  being  finite  and  measurable  (much  as 
the  weights  of  atoms  are  measurable),  in  spite  of  the  smallness 
of  the  terms.  This  view  is  no  longer  held.  The  expression 

-^  is  not  a  ratio  at  all  but  only  the  limit  which  the  ratio 
dx 

of  the  increases  of  the  variables  approaches  as  the  increment 
of  x  approaches  zero.  Naturally  the  learner  is  greatly  puzzled 


4:48  CRAFTSMANSHIP   IN  MATHEMATICS 

if  he  is  told  to  write  the  derivative  in  the  form  of  a 
fraction  and  is  then  forbidden  to  think  of  it  as  a  fraction. 
Thus  it  is  much  the  best  plan  to  withhold  the  Leibnizian 
notation  at  first.  Use  the  symbol  D(y)  instead;  this  symbol 
reminds  the  pupil  that  he  is  seeking  a  function  which  he  is 
to  derive  from  the  given  function  y  by  means  of  a  definite 
rule  of  procedure.  This  relationship  between  functions  is 
the  essence  of  the  whole  matter. 


Integration. — Like  current  ideas  about  the  nature  of  a 
differential  coefficient,  those  about  the  nature  of  an  integral 
also  show  traces  of  the  erroneous  mathematical  philosophy 
of  earlier  days.  The  problems  first  systematically  studied  by 
Wallis  came  to  be  regarded  as  having  for  their  aim  the 
summation  of  an  "  infinite  "  number  of  "  infinitesimals  "  dy, 
of  the  form  y.dx.  This  view  is  still  represented  not  only  by 
the  usual  notation  I  =  \y.dx,  which  (like  dy/dx)  was  in- 
troduced by  Leibniz,  but  also  by  the  common  statement 
that  an  integral  is  the  sum  of  an  infinite  number  of  infinitely 
small  magnitudes.  With  the  rejection  of  the  notion  of  an 
infinitesimal  as  a  definite  atomic  magnitude,  this  statement, 
and  the  notation  which  expresses  it,  have  become  inadmissible. 
If  dx  has  any  numerical  significance  at  all,  it  stands  for  the 
increment  h  when  h  is  zero.  Hence  the  product  y.dx  is  also 
zero  for  all  values  of  y,  and  therefore  the  summation  repre- 
sented by  jy  .  dx  is  the  summation  of  a  series  of  zeros!  I  is 
not  the  sum  of  an  infinite  number  of  products;  *  it  is 
simply  the  limit  of  the  sum  of  a  finite  number  of  pro- 
ducts. 

There  is  neither  need  nor  warrant  for  introducing  the 
term  "  infinite  "  at  any  point  of  the  discussion.  If  we  sub- 
stitute the  useful  A#  for  the  absurd  dxy  we  may  still  usefully 
retain  the  Leibnizian  mode  of  expression  I  =  fy  .  A#,  but  the 
symbol  "  /  "  must  now  be  read,  "  limit  of  the  sum  as  A# 
approaches  zero  ". 


THE  CALCULUS:    SOME  FUNDAMENTALS       449 
Interpretation  of  -¥. 

n    Ay       distance  j  j     •       A 

1.  -r^-  =  — : =  average  speed  during  A#; 

A*          time  *      F  fe 

dy limit   of  average  speed  =  "  instantaneous  " 

d~x ~~      speed. 

2.  -—  —  average  slope  of  curve  during  interval  A#; 

L\X 

j-  =  limit  of  average  slope  =  limit  at  point  P. 

The  two  problems  (1)  to  determine  the  rate  of  increase 
of  a  function  and  (2)  to  draw  a  tangent  to  a  curve,  are  really 
identical;  if  we  have  a  general  method  of  determining  the 
rate  of  increase  of  a  function  f(x)  of  a  variable  x,  we  are 
able  to  determine  the  slope  of  the  tangent  at  any  point  (x,  y) 
on  the  curve. 

Points  for  emphasis. — We  will  once  more  stress  the  points 
to  be  kept  in  the  forefront  of  the  teaching. 

The  pupils  should  be  told  plainly  that  the  old  idea  of 
infinitely  small  quantities  has  been  definitely  abandoned. 
The  real  explanation  of  the  whole  thing  was  first  put  forward 
by  a  German  mathematician,  Weierstrass,  about  the  middle 
of  the  nineteenth  century. — The  subject  had  been  sound 
enough;  so,  virtually,  had  been  the  mathematical  procedure, 
but  the  explanation  had  been  wrong. 

The  problem  was  to  retain  an  interval  of  length  A,  over 
which  to  calculate  the  average  increase,  and  at  the  same  time 
to  treat  h  as  if  it  were  zero.  As  Professor  Whitehead  says, 
"  As  long  as  we  look  upon  '  h  tending  to  a  '  as  a  fundamental 
idea,  we  are  in  the  clutches  of  the  infinitely  small,  for  we 
imply  the  notion  of  h  being  infinitely  near  to  a.  This  is 
what  we  want  to  get  rid  of."  "  The  limit  of  f(h)  at  a  is  a 
property  of  the  neighbourhood  of  a."  "  In  finding  the  limit 

(E291)  30 


450  CRAFTSMANSHIP  IN  MATHEMATICS 

Z»/Ov         I          Jj\ 

of  -^ — 1  at  the  value  0  of  the  argument  hy  the  value 

h 

(if  any)  of  the  function  at  h  =  0  is  excluded.  But  for  all  values 
of  h  except  h  =  0  we  can  divide  through  by  h."  "  In  the 
neighbourhood  of  the  value  0  for  h,  2x  +  h  approximates 
to  2x  within  every  standard  of  approximation,  and  there- 
fore 2x  is  the  limit  of  2x  +  h  at  h  =  0.  Hence,  at  the  value 

0  for  h,  2x  is  the  limit  of  (*  +  hY  ~  **» 

h 

The  difficulty  of  former  mathematicians  was  that  on  the 
one  hand  they  had  to  use  an  interval  h  over  which  to  calculate 
the  average  increase,  and  on  the  other  hand  they  wanted  to 
put  h  =  0.  "  Thus  they  seemed  to  land  themselves  with 
the  notion  of  an  existent  interval  of  zero  size."  Present-day 
mathematicians  avoid  that  difficulty  by  using  the  notion 
that,  corresponding  to  any  and  every  possible  standard 
of  approximation,  there  is  still  some  interval. 

My  own  experience  is  that  when  Sixth  Form  boys  are 
puzzled  over  this  question,  their  puzzlement  is  almost  always 
due  to  the  fact  that  they  have  got  hold  of  the  term  infinity, 
and  do  not  understand  what  the  term  signifies. 

Books  to  consult: 

1.  The  Teaching  of  Algebra ,  Nunn. 

2.  An  Elementary  Treatise  on  the  Calculus,  Gibson. 

3.  Course  of  Pure  Mathematics,  Hardy. 

4.  Applied  Calculus,  Bisacre.     (An  outstanding  book.) 


WAVE  MOTION  461 

CHAPTER  XXXI 

Wave   Motion:    Harmonic  Analysis: 
Towards   Fourier 

Sine  and  Cosine  Curves.     Composition 

The  pupils  will,  of  course,  be  thoroughly  familiar  by 
this  time  with  the  radian  notation,  and  will  understand  that 
the  reason  for  measuring  angles  in  radians  is  that  theoretical 
arguments  are  simplified.  They  will  know  that  TT  radians=  180°; 
that  as  an  angle  of  9  radians  is  subtended  by  an  arc  of  Or, 
the  length  of  an  arc  of  a  circle  =  r9\  that  the  symbols  6  and 
(/>  are  commonly  used  in  circular  measure,  and  the  symbols 
a,  /?,  y  for  measurements  in  degrees.  They  ought  also  to 
know  that,  when  an  angle  is  small,  its  circular  measure  may, 
in  approximation  calculations,  be  substituted  for  its  sine 
(or  tangent);  and  that,  when  it  is  not  small,  the  values  of 
the  sine  and  cosine  may  still  be  expressed  approximately  in 
circular  measure  by  means  of  the  simple  formulae  sin  9  = 

03  92 

0  —  — ,  cos0  =  1  —  — .    The  proofs  of  these  may  be  given  at 
6  2 

an  appropriate  stage,  but  a  simple  graphic  method  is  easily 
devised  to  suggest  that  the  formulae  are  approximately  true. 

When  boys  are  first  introduced  to  angles  greater  than 
360°,  they  are  inclined  to  doubt  if  they  are  dealing  with  real 
things,  and  to  be  a  little  sceptical  about  the  practical  value 
of  the  work  in  hand.  Light,  however,  begins  to  dawn  when 
they  are  introduced  to  Simple  Harmonic  Motion,  to  waves, 
and  to  spirals. 

They  must  be  made  to  understand  clearly  that  the  values 
of  the  ratios  connected  with  an  angle  are  repeated  endlessly 
in  cycles  as  the  angle  rises. 

They  will,  of  course,  be  thoroughly  familiar  with  the  sine 
and  cosine  curves.  With  very  little  practice  they  can  make 


452 


CRAFTSMANSHIP  IN  MATHEMATICS 


a  supply  of  these  curves  for  themselves  by  running  them 
off  from  Fletcher's  trolley  arranged  for  uniform  motion; 
with  care,  these  curves  may  be  obtained  to  a  surprising 
degree  of  accuracy.  Draw  tangents  to  the  succession  of 
crests,  then  the  axis  midway  between  them.  The  chief 
ordinates  are  the  perpendiculars  at  those  points  of  the  axis 
midway  between  the  nodes. 

Point   out   that   all   sine   curves   have  the  same   general 
shape  and  properties;  that  smx  gives  a  wave  curve  of  period 

Y 


2?r  with  successive  maximum  and  minimum  values  at  +1 

and  —  1  respectively;  that  sinpx  gives  a  wave  curve  of  period 
o  o 

— ;  that  asin(px-\-e)  gives  a  wave  curve  of  period  — , 
P  P 

with  successive  maximum  and  minimum  values  of  -\-a  and 
—a,  respectively,  the  effect  of  e  being  merely  to  displace 
the  curve  along  the  axis. — These  fundamentals  must  be 
mastered.  The  py  the  ay  and  the  e  are  veritable  traps  for 
the  unwary  beginner;  the  inner  significance  of  the  three 
symbols  should  be  expounded  and  emphasized  again  and 
again. 

From  his  earlier  knowledge  of  graphs,  a  boy  may,  without 
further  instruction,  graph  one  or  two  easy  cases  of  compound 
periodic  functions.  We  give  two  examples  adapted  from 
examples  in  Siddons  and  Hughes'  Trigonometry,  the  first, 
2  sin*  +  3  cos#,  consisting  of  two  periodic  functions  of 
the  same  period  (fig.  256),  and  the  second  sin3#  +  2  sin*, 
periodic  functions  of  different  periods  (fig.  257).  In  each 
case  the  two  functions  are  first  plotted  separately  (the  curves 
are  shown  by  lighter  lines),  then  the  required  composite 


WAVE  MOTION 


453 


curve  is  obtained  by  means  of  points  determined  by  taking 
the  algebraic  sum  of  the  ordinates  of  the  constituent  curves. 

Y 

4 


1'ig.  256 


For  instance,  in  the  second  case  pm  =  pn  +  pq.     Note  in 
the  first  case  where  we  are  compounding  functions  of  the 
Y 


\ 


^ 

Fig.  257 


period,  the  result  is  a  «'«e  curve;    in  the  second  case, 
where  the  functions  to  be  compounded  are  not  of  the  same 


454  CRAFTSMANSHIP  IN  MATHEMATICS 

period,  the  result,  though  a  periodic  curve,  is  not  a  sine 
curve.  If  in  the  second  case  we  slide  the  half  curve  TT  to  2ir 
along  the  x  axis  up  to  the  y  axis,  the  upper  and  lower  halves 
will  easily  be  seen  to  be  symmetrical.  Observe  that  in  the 


case  of  sin  3x  the  period  is  -  =  120°. 

<j 

In  all  such  cases  the  shape  of  tlje  composite  curve  can 
be  obtained  only  by  plotting  a  wide  range  of  ordinate  values, 
though  this  is  always  simply  done  by  algebraic  addition, 
and  a  pair  of  dividers  will  soon  give  the  necessary  number 
of  points.  The  curves  are  a  little  tricky  to  draw,  because 
of  the  minus  quantities  to  be  added. 

This  kind  of  exercise  need  take  but  little  time.  I  have 
known  boys  work  through  half  a  dozen  in  an  hour.  The 
general  shapes  of  the  sine  and  cosine  curves  are  already 
familiar,  and  as  the  axis  can  be  divided  up  and  the  principal 
ordinate  put  in  at  once,  the  constituent  curves  can  be  sketched 
in  in  less  than  a  minute.  It  is  assumed,  of  course,  that  the 
significance  of  p,  a,  and  e  has  been  fully  grasped.  But  the 
negative  quantities  to  be  considered  when  building  up  the 
composite  curve  nearly  always  give  trouble.  No  calculations 
are,  however,  necessary.  Let  the  dividers  do  the  work, 
unless,  in  some  very  exceptional  instances,  rigorous  accuracy 
is  wanted.  The  general  form  and  what  it  teaches  is  the 
main  thing. 

Waves  and  their  Production 

Since,  in  most  instances,  waves  are  periodic  phenomena, 
they  afford  excellent  concrete  examples  of  periodic  functions. 
No  one  can  appreciate  the  most  striking  triumphs  of  physical 
science  who  has  not  given  some  attention  to  the  mathematics 
of  wave  motion.  In  fact,  wave  motion  now  forms  the  very 
basis  of  the  study  of  the  greater  part  of  physics,  and,  after 
all,  the  necessary  mathematics  of  the  subject  is,  in  all  its 
main  factors,  quite  simple.  It  need  hardly  be  said  that 
that  new  and  rather  formidable  subject,  Wave  Mechanics, 
is  outside  the  scope  of  school  practice. 


WAVE  MOTION  455 

The  teaching  of  such  characteristics  of  waves  as  resistance, 
persistence,  and  over-shooting  the  mark,  is  part  of  the  business 
of  the  physics  master.  The  mathematical  master  is  concerned 
mainly  with  considerations  of  the  form  of  the  wave  and  its 
analysis.  Let  beginners  first  read  through  Fleming's  Waves 
and  Ripples,  and  so  supplement  the  work  they  have  already 
done  in  the  physics  laboratory;  the  mathematics  will  then 
give  them  little  trouble.  But,  if  they  begin  the  mathematics 
of  wave  motion  before  they  have  acquired  in  the  laboratory 
a  considerable  amount  of  practical  knowledge  of  the  subject, 
they  will  never  be  quite  sure  of  their  ground. 

The  boys  will  probably  be  familiar  with  the  device  of 
producing  a  train  of  waves  by  means  of  a  length  of  narrow 
stair  carpet,  or  a  sand-filled  length  of  rubber  tubing,  or  a 
length  of  heavy  rope:  these  things  are  part  of  the  stock 
in  trade  for  teaching  wave  motion  in  the  physics  laboratory. 
An  instructive  experiment  is  the  following:  take  a  common 
blind-roller  about  5'  long,  with  a  pulley  runner  fixed  at  each 
end.  Into  the  roller  drive  37  4"  nails,  at  1J"  intervals,  in 
the  form  of  a  uniform  spiral  of  3  complete  turns.  The  nails 
should  be  separated  from  one  another  by  a  uniform  interval 
of  30°,  so  that  the  1st,  13th,  25th,  and  37th  are  in  the  same 
straight  line;  the  2nd,  14th,  and  26th  in  another  straight 
line;  and  so  on.  Support  the  roller  in  a  horizontal  position 
in  front  of  a  white  screen,  and  turn  it  by  means  of  an  im- 
provised crank.  Let  a  distant  light  throw  on  the  screen 
a  shadow  of  the  rotating  roller.  Observe  how  the  shadows 
of  the  nail-heads  exhibit  progressive  wave  motion.  Observe 
the  movement  of  any  one  particular  shadow;  it  is  an  example 
of  simple  harmonic  motion  (see  Chapter  XXXVII).  The 
travelling  shadow-wave,  constituted  by  equal  simple  harmonic 
motions  of  the  shadows  of  the  nail-heads,  is  a  progressive 
harmonic  wave.  The  shadow  of  any  head  differs  in  phase 
from  that  of  its  neighbours  by  a  constant  amount  of  30°. 
Note  that  each  nail  remains  in  its  own  vertical  plane;  the 
progressive  horizontal  movement  is  one  of  form  only.  The 
boys  must  distinguish  between  (1)  the  actual  to  and  fro 


456 


CRAFTSMANSHIP  IN  MATHEMATICS 


movements  of  elements  in  a  wave-medium,  and  (2)  the  move- 
ment of  the  wave  itself.  The  second  is  merely  an  appearance, 
resulting  from  the  successive  movements  of  the  first.  The 
first  has  the  effect  of  making  successive  sections  of  the  medium 
(as  we  may  conveniently  call  it)  assume  one  after  another 
the  same  shape.  The  shape  therefore  appears  to  be  some- 
thing moving  along. 

The  waves  on  the  surface  of  the  sea,  away  from  the  shore, 
are  good  examples  of  progressive  waves.  If  their  outline 
were  exactly  a  sine  curve,  as  theoretically  it  should  be,  we 
should  have  an  example  of  a  harmonic  progressive  wave. 


Common  Wave  Formulae 


In  figure  258, 


the  wave-length  —  LtL2  =  L2L3  =  A, 
the  amplitude     =  PQ  =  a. 


I x 


Fig.  258 

If  T  is  the  periodic  time  of  the  wave  (i.e.  time  to  complete 
a  vibration),  it  follows  from  first  principles  that  v  =  A/T  =  «A, 
where  v  =  velocity  and  n  —  frequency. 

Let  Lj#  be  d\  let  pq  be  h.   Since  the  curve  is  a  sine  curve, 

A  =  360°  =  2?r.   Hence  the  number  of  degrees  in  d  =  —  X  d. 

277  A 

For  -T-  write  p.    Then,  wherever  q  is  taken  along  L^Lo,  it 

A 

may  be  found,  by  actual  measurement,  that  h  =  a  sin£d. 
This  is  a  fundamental  formula. 

Let  y  =  a  sinpx  describe  the  wave  outline  in  a  given 
position,  x  being  measured  from  Lj.  If  the  curve  move  to  the 


WAVE  MOTION  457 

right  with  a  velocity  v,  its  form  after  t  seconds  is  given  by 
the  formula, 

y  —  a  s'mp(x  —  vt) (i) 

If  to  the  left  with  the  same  velocity, 

y  =  a  sinp(x  +  vt) (ii) 

These  formulae  are  simply  applications  of  the  general  prin- 
ciple that  if  a  graph  is  moved  a  distance  d  parallel  to  the 
x  axis,  (x  —  d)  must  be  substituted  for  x  in  the  formula. 

Since  p  —  — ,  (i)  and  (ii)  may  be  expressed  thus: 
A 

y  =  a  sin  —  (x  ±  vt). 

A 

The  actual  significance,  in  the  graph,  of  each  symbol  in 
this  formula  must  be  understood. 

Compound  Harmonic  Waves 

Let  two  boys  near  each  other  on  the  edge  of  a  pond, 
or  other  suitable  sheet  of  still  water,  each  produce  a  series 
of  waves  by  striking  the  water  rhythmically  with  a  stick. 
Let  the  frequency  of  the  blows  be  2  to  3  (say  2  in  2  seconds 
and  3  in  2  seconds,  easily  done  after  a  little  practice  with 
watch  in  hand),  and  suppose  the  waves  to  travel  with  the 
velocity  v.  A  pattern  will  result  from  the  five  waves  which 
are  produced  every  two  seconds,  and  it  will  be  regularly 
repeated,  though  gradually  fading  away  into  ripples.  But 
this  pattern  will  no  longer  represent  simple  harmonic  waves, 
for  the  shape  which  appears  to  move  along  the  water  beyond 
the  ends  of  the  line  joining  the  centres  of  disturbance,  is 
no  longer  a  simple  sine  wave;  the  length,  the  frequency, 
and  the  amplitude  of  the  resultant  waves  will  be  different 
from  the  length,  the  frequency,  and  the  amplitude  of  the 
component  waves.  At  points  reached  simultaneously  by 
crests  and  troughs  belonging  to  the  component  wave-trains, 


458  CRAFTSMANSHIP  IN  MATHEMATICS 

the  elevation  or  depression  of  the  surface  is  exaggerated. — 
All  this  should  be  confirmed  by  observation. 

To  calculate  the  resultant  disturbance  due  to  the  two 
component  waves  (assumed  to  be  simple  harmonic  waves), 
we  adopt  the  principle,  which  accords  with  observation, 
that  the  actual  displacement  at  any  point  is  equal  to  the 
algebraic  sum  of  the  displacements  due  to  the  waves  separately. 

If  the  first  wave-train  existed  alone,  the  displacements 
produced  would  be  represented  by  moving  the  curve 

y  =  a:  sin—  x  with   velocity  v   towards   the   right.      If  the 

^i 
second  wave-train  existed  alone,  the  displacement  produced 

would  be  represented  by  moving  the  curve  y~a2s'm—- (x — c) 

2 
with  velocity  v  towards  the  right.    Here,  c  is  the  x  co-ordinate 

(at  t  =  0)  of  the  nearest  point  of  the  wave  from  the  centre 
of  disturbance,  comparable  with  L  in  fig.  258. 

Thus  the  actual  character  of  the  resultant  composite 
wave  is  represented  by  the  graph 

.   2ru      ,          .    27U,          x 
y  =  al  sin  —  x  -j-  a2  sin  —  (x  —  c), 
AJ  X2 

moving  to  the  right  with  a  velocity  v. 

This  evaluation  from  first  principles  is  really  very  simple, 
but,  unless  it  is  associated  with  at  least  a  little  experimental 
work,  it  may  prove  difficult  for  average  boys.  The  formula 
is  a  key  formula  and  should  be  mastered. 

Had  there  been  three  boys  at  the  pond  side,  each  pro- 
ducing waves  by  striking  the  water  rhythmically,  all  the 
waves  being  of  different  length,  the  composite  waves  would 
have  been  more  complex,  and  the  necessary  formula  for  the 
graph  would  have  consisted  of  three  terms.  So  generally. 


WAVE  MOTION  459 

Comparison  of  Periodic  and  non -Periodic 
Functions 

Functions  of  the  form  s'mpx  and  cospx  (where  p  =  ~\ 

\  A  / 

have  much  the  same  relation  to  periodic  curves  as  x  has  to 
non-periodic  curves.  The  simplest  non-periodic  curve  is 
the  straight  line  y  —  Ax  (we  write  it  in  different  forms, 
according  to  circumstances;  e.g.  y  —  mx  -f-  c);  and  the 
simplest  periodic  curve  is  y  =  A  s'mpx  (also  written  in 
different  forms  according  to  circumstances). 

With  our  former  non-periodic  work  we  soon  learnt  that 
the  curve  y  =  AI(*  +A2#2  was  more  complex  than  y  =  Ax, 
and  that  y  —  A^x  +  A2x2  +  A3x?  was  more  complex  still; 
and  so  on;  a  quadratic  function  was  more  complex  than 
a  straight-line  function,  a  cubic  more  complex  than  a 
quadratic.  Still,  however  complex  the  function,  it  was 
always  a  question  of  the  addition  of  a  number  of  terms; 
the  actual  graphing  was  simple  enough  though  tedious  if 
the  terms  were  many. 

So  it  is  with  periodic  functions,  where  the  curve,  whether 
simple  or  complex,  recurs  endlessly,  and  makes  a  continuous 
wave.  The  effect  of  adding  to  y  =  A  s'mpx  the  term  A2  sin  2px 
may  be  compared  with  that  of  adding  to  y  —  A*  the  term 
A2#2;  in  each  case  we  obtain  a  form  of  greater  complexity. 
By  adding  further  terms  we  get,  in  each  case,  still  further 
complexity,  save  that  in  the  former  case  the  successive  curves 

all  have  the  period  A  —  —  or  a  submultiple  of  this. 

P 
The  standard  form  of  a  periodic  function  may  be  written: 

y  —  Aj  s'mpx  +  A2  s'mSpx  +  A3  sin3j>#  +   .  .  .  -f  Ar  s'mrpx. 

It  was  the  French  mathematician  Fourier  who  first  observed 
that  a  periodic  function  of  unlimited  complexity  may  be 
described  by  a  formula  of  this  type.  The  process  of  deter- 
mining the  components  of  which  a  given  periodic  function 


460 


CRAFTSMANSHIP  IN  MATHEMATICS 


is  the  resultant  is  known  as  harmonic  analysis.     Fourier's 
statement  is  known  as  Fourier's  theorem. 

Let  the  boys  consider  an  illustration  of  this  kind.  Let 
them  inagine  a  water  wave  sent  out  with  a  velocity  v,  of 
length  A,  and  frequency  1  per  second;  the  wave  would  form 
a  simple  sine  curve,  such  as  we  see  on  any  disturbed  water 
surface.  Now  let  them  imagine  a  second  wave,  sent  out  from 
the  same  point,  independently  but  at  the  same  moment,  at 
a  frequency  of  2  per  second,  with  the  same  velocity  v  and 
therefore  of  a  wave-length  A/2.  This  second  wave  would 
not  have  the  appearance  of  its  independent  self  but  would 
be  imposed  on  the  other,  and  what  we  should  see  travelling 
along  the  water  surface  would  be  a  composite  wave.  Now 
let  them  imagine  a  third  wave  to  be  sent  out  from  the  same 
point,  independently  but  at  the  same  moment  as  the  other 
two,  at  a  frequency  of  3  per  second,  with  the  same  velocity 
v,  and  therefore  of  a  wave-length  A/3.  (Remember  that 
v  =  nXy  always.)  This  third  wave  will  not,  any  more  than 
the  second,  show  itself  independently;  it  will  simply  make 
the  previous  composite  wave  still  more  complex.  And  so 
we  might  go  on.  The  waves  sent  out  independently  might 
be  shown  thus: 


[-,  \( 


/ 

• 

ot 

is 

/ 

s 

M 

s 

s 

S 

/ 

s 

/ 

Zir 

k 

/ 

—       xi        - 

/ 

"\ 

/ 

\ 

/ 

\ 

M 

/ 

\ 

/ 

s 

s 

2. 

\ 

/ 

\ 

/ 

Zir 

\ 

/ 

—  -X—  — 

Fig.  259 


WAVE  MOTION 


461 


All  3  waves  start  from  O,  and  since  they  travel  with  the 
same  velocity  they  reach  M  at  the  same  moment,  but  by 
that  time  the  first  will  have  completed  1  of  its  periods,  the 
second  2,  the  third  3.  Imagine  a  whole  series  of  waves  sent 
out  in  this  way,  each  of  them  with  a  wave-length  which  is 
a  submultiple  of  A,  though  not  necessarily  all  the  members 
of  the  sequence  A,  A/2,  A/3,  A/4,  &c.:  some  of  the  series  may 
be  missing.  Now  imagine  the  water  to  be  suddenly  frozen, 
so  that  the  wave  would  be  set  in  ice,  and  its  section  readily 
drawn.  We  might  have  a  composite  wave  like  fig.  260, 
OM  composing  a  unit  which  would  be  repeated  endlessly 
until  the  wave  died  away  in  a  ripple.  The  problem  is,  how 


/ 

\ 

^\ 

/ 

\ 

r\  . 

V 

y 

\^ 

/ 

\ 

J 

\J 

^  x  ^ 

Fig.  260 

are  we  to  analyse  this  curve,  in  order  to  discover  all  the 
simple  curves  of  which  it  is  compounded. 

The  waves  need  not  all  have  been  sent  out  with  the  same 
amplitude  (<z),  as  shown  in  fig.  259.  Neither  need  they  have 
been  sent  out  from  exactly  the  same  point;  one  might  have 
been  started  40°  or  50°  farther  along  the  axis  than  the  others. 
And  remember  that  a  cosine  curve  is  produced  from  a  sine 
curve  merely  by  pushing  it  forwards  or  backwards  90°  along 
the  axis.  We  may  therefore  easily  find  cosines  as  well  as 
sines  in  our  formulae;  it  is  all  a  question  of  convenience, 
depending  on  the  particular  curves  under  consideration. 

Briefly,  Fourier's  statement  was  this:  Any  repeated 
complex  wave  pattern  of  length  A  may  be  produced  by  adding 
to  a  certain  fundamental  sine  or  cosine  curve  of  length  A, 
sine  or  cosine  curves  of  the  proper  amplitudes  whose  lengths 
are  A/2,  A/3,  A/4,  &c.  Conversely,  the  complex  pattern  may 
be  revolved  into  its  original  component  sine  and  cosine 


462  CRAFTSMANSHIP  IN  MATHEMATICS 

curves,  since  any  of  the  unknown  amplitudes  may  be  deter- 
mined at  will. 

Observe  that  the  main  difficulty  in  analysing  the  complex 
curve  arises  from  the  fact  that  the  component  curve  may  be 
of  different  amplitudes.  The  general  expression  for  the 
complex  curve  may  be  written  in  different  ways,  though 
they  all  mean  the  same  thing. 

(i)  y  =  a0  +  (#1  sin  px  +  bl  cos  px) 

+  (a2  sin2/>#  +  b2  cos2/>#)  +     .  .  . 
(ii)  y  =  a0  +  a±  sin(*  +  «i)  +  #2  sin(2#  +  a2) 

+  a3  sin(3#  +  a3)+  .  .  . 
(iii)  y  =  a0  +  a  sin(0  +  a)  +  b  sin(20  +  j3) 
+c  sin(30 


Remember  that  p  =  2?7/A. 

The  constant  a0  meets  the  case  in  which  the  x  axis  is 
not  identical  with  the  common  axis  of  the  various  harmonic 
curves. 

Observe  that  each  term  in  the  above  expressions  represents 
a  simple  harmonic  function.  Those  harmonics  in  which  the 
coefficient  of  x  is  an  odd  number  are  called  odd  harmonics; 
those  in  which  the  coefficient  of  x  is  even  are  called  even 
harmonics.  Observe,  too,  that  the  second  term  gives  a  curve 
with  twice  as  many  complete  waves,  the  third  term  a  curve 
with  three  times  as  many  complete  waves  (and  so  on),  as  the 
first  or  fundamental  term.  This  is  exemplified  in  fig.  259, 
where  the  "  period  "  of  the  second  term  is  \  the  period  of 
the  first,  the  period  of  the  third  is  \  the  period  of  the  first; 
and  so  on.  The  frequencies  are  therefore  twice,  three'  times, 
&c.,  the  frequency  of  the  first. 

Curve  Composition 

Let  us  first  consider  curve  composition.  It  is  very  simple- 
Plot  to  the  same  axis  the  successive  components,  al  sin(#+ai)» 
az  sin(2#  +  a2),  &c.,  and  then  add  the  corresponding  ordinates 
to  obtain  the  respective  ordinates  of  the  composite  curve 


WAVE  MOTION 


463 


(cf.  figs.  256,  257).     Since  the  first  or  fundamental  term  is 
represented  by  the  period  0  to  2?r,  the  wave  will  consist  of 
repetitions  of  the  first  portion  between  x  =  0°  and  x  =  360°. 
We  give  the  graph  of 

100  sin*  +  50  sin(3*  —  40°), 

from  0°  to  360°.     The  first  or  fundamental  term,  100  sin  x, 
represents  the  first  harmonic  with  an  amplitude  100  (=al  in 


100 


50. 


Fig.  261 

general  formula).  The  second  term,  50  sin(3#  —  40°),  is 
the  third  harmonic  with  an  amplitude  of  50  (=a2),  and 
consists  of  three  complete  waves  within  the  period  of  the 
first. 

The  function  consists  of  only  odd  harmonics,  and  in 
virtue  of  this  fact  the  graph  possesses  a  special  kind  of 
symmetry  characteristic  of  all  curves  containing  only  odd 
harmonics.  If  the  portion  of  the  graph  from  n  to  2?r  be  made 


464 


CRAFTSMANSHIP  IN  MATHEMATICS 


to  slide  to  the  left,  to  the  position  0  to  TT,  it  will  be  the  re- 
flected image  of  the  half  above  the  axis.  (Cf.  fig.  257.)  Note 
that  the  composite  curve  is  not  a  sine  curve. 

Had  the  function  contained  the  absolute  term  a0,  say 

y  =  70  +  100  sin  x  +  50  sin(3*  -  40°), 

the  graph  would  be  the  same  as  before  but  raised  vertically 
70  units.  The  line  of  symmetry  referred  to  above  would  then 
no  longer  be  the  x  axis. 

We  give  a  second  example,  this  time  consisting  of  the 
first  and  second  harmonics: 

y  =  10  roi(6  +  30°)  +  5  sin(20  +  45°). 

We  will  plot  the  graph  from  a  tabulated  series  of  values, 
though  this  is  really  unnecessary. 

If  10sin(«  +  30°)=<y1,  and  5  sin  (20  +  45°)  =  y2,  then 


when  0  =  0°,  yi  =  10  sin  30°  =  5,  and  y2  =  5  sin  45°  = 

i 

also  when  6  =  30°,  yl  =  10  sin  60°  =  8-66  and  y2  =  5  sin  105° 
=  5  sin  75°  =  4-8.    Similarly  other  values  may  be  calculated. 


Note  the  device  of  running  off  horizontals  from  a  graduated 
circle.  Since  the  "  period  "  in  the  x  axis  is  divided  into  12 
equal  parts,  we  divide  the  circumference  of  the  circle  also 
into  12  equal  parts. 


WAVE  MOTION 


465 


Values  ^ 
of  6    f 

0°. 

30°. 

60°. 

90°. 

120°. 

y\  = 

5 

8-65 

10 

8-65 

6 

y2  = 

3-5 

4-8 

1-3 

-3-5 

-4-8 

yi  +  y*  = 

8-5 

13-45 

11-3 

5-15 

0-2 

Observe  that  as  the  point  on  the  smaller  circle  rotates  at 
twice  the  rate  of  a  point  on  the  larger,  it  is  only  necessary  to 
divide  the  smaller  circle  into  half  as  many  parts  as  the  larger. 
Set  up,  say,  12  ordinates  for  the  whole  line  0°  to  360°,  then 
divide  the  circumference  of  the  larger  circle  also  into  12 
parts,  and  run  off  parallels  to  cut  the  ordinates.  Each  second 
harmonic  will  embrace  only  6  of  the  12  ordinates,  and  hence 
only  6  parallels  from  the  smaller  circle  are  required.  The 
radii  of  the  circles  are,  of  course,  equal  to  the  amplitudes 
of  the  respective  harmonics.  Observe  the  plan  for  fixing 
the  first  point  of  each  harmonic. 

Functions  with  more  terms  than  two  are  treated  in  exactly 
the  same  way,  but  naturally  the  composition  is  a  tedious 
operation. 

Curve  Analysis 

Secondly,  we  come  to  the  decomposition  or  analysis  of 
a  composite  curve.  This  is  much  less  simple  than  the  reverse 
operation. 

The  composite  curve  may  be  the  resultant  of  two  or  more, 
perhaps  a  large  number,  of  harmonics.  But  it  does  not  at 
all  follow  that,  because  a  particular  harmonic,  say  the  ninth, 
has  been  included  in  the  building  up,  therefore  all  the  earlier 
ones  (in  this  case  the  first  8)  of  the  series  are  included  too. 
How  are  we  to  discover  which  harmonics  are  included,  and 
how  are  we  to  draw  them? 

Whatever  scheme  we  adopt,  it  is  advisable,  when  we  have 
discovered  the  component  harmonics,  to  draw  them  all 
carefully,  to  compound  them  again,  and  to  see  if  the  result 
corresponds  to  the  original  curve. 

(£291)  31 


466 


CRAFTSMANSHIP  IN  MATHEMATICS 


Let  integration  wait  until  a  later  stage.  Let  the  boys 
first  learn  what  the  new  thing  is  really  about.  Let  them 
consider  the  few  simple  cases  which  may  easily  be  solved 
graphically,  and  after  all,  these  are  the  cases  of  greatest 
practical  importance  (for  instance  those  that  are  concerned 
with  the  theory  of  alternating  currents).  Such  cases  may, 
with  sufficient  approximation,  be  represented  by  the  sum  of 
two  or  three  harmonic  terms. 

We  select  as  an  example  one  of  Mr.  Frank  Castle's 
engineering  problems.  The  curve  in  the  figure  is  drawn 


'% 

^° 

1* 

I 

vz 

j^"  ^^Sk 

_jr                  IV 

7      [\ 

O      1        2       3       4\ 

y_ 

6 

7 

G 

9 

O      \\     712 

\ 

/ 

> 

\ 

V 



N 

\ 

/ 

through  12  successive  positions  of  a  slide  valve,  corresponding 
to  intervals  of  30°  of  the  crank,  beginning  at  the  inner  dead 
point.  It  is  required  to  analyse  the  motion  so  as  to  express, 
in  the  form  of  a  series  of  harmonics,  the  displacement  of  the 
valve  from  its  mean  position.  (Practical  Mathematics y  p.  459.) 

Were  the  curve  divided  more  symmetrically  by  the  x 
axis,  we  should  suspect  comparatively  little  deviation  from 
the  first  harmonic,  i.e.  an  ordinary  sine  curve.  But,  fairly 
obviously,  it  is  compounded  with  other  harmonics  as  well. 

Run  off  the  lengths  of  the  ordinates  to  the  edge  of  a 
paper  strip  as  shown  in  fig.  263.  Use  the  strip  for  plotting 
the  points  in  fig.  264,  but  first  reverse  it,  so  that  point  8 
is  at  the  top  and  point  2  at  the  bottom. 


WAVE  MOTION 


467 


For  the  first  harmonic. — Let  0  of  the  strip  coincide  with  O 
in  fig.  264,  and  mark  off  these  distances:  0  to  6  on  the  ordinate 
through  O,  1  to  7  on  the  ordinate  through  1,  2  to  8  on  the 
ordinate  through  2,  3  to  9  on  3,  4  to  10  on  4,  5  to  11  on  5  (six 
measurements  in  all).  Observe  that  these  distances  on  the 


Fig.  264 

strip  give,  successively,  yQ  -  y^  y1—  y7,  y2  —  j;8,  y3  —  yg, 
y±  ~~  Jio>  Vb  ~~"  Vii-  Draw  a  curve  through  the  points,  then 
the  second  half  of  the  curve,  below,  through  points  obtained 
from  the  same  measurements,  reversed.  Draw  a  tangent  to 
this  curve  at  a  maximum  or  minimum  point,  HK,  MN. 
The  amplitude  a'  is  half  the  distance  from  the  tangent  to 
the  axis;  it  is  7/2  =  3-5. 


468 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  magnitude  of  the  angle  a  can  be  obtained  by  measuring 
the  length  Op.  The  distance  O  to  6  =  180°;  hence  Op  =  151-8°. 
Thus  a  =  180°  -  151-8°  -  28-2°. 

For  the  second  harmonic. — The  successive  distances  for 
the  ordinates  to  be  taken  from  the  strip  are  (0  to  3)  +  (6  to  9), 
(1  to  4)  +  (7  to  10),  (2  to  5)  +  (8  to  11).  Observe  that  these 
distances  are  (y0  -  y3)  +  (y6  -  JVg),  CVi  —  Vi)  +  Cv?  ~  J>io)> 
(yz  ~~  y&)  +  (y&  ~~  y\\)>  Draw  a  curve  through  the  points, 
and  repeat  below;  and  do  the  same  thing  again  for  the 
second  period  of  the  wave. 

To  obtain  the  amplitude  a",  draw  a  tangent,  and  take 
J  of  the  distance  to  the  x  axis,  =  -25.  To  obtain  the  angle 
£,  measure  O?;  Oq  -  £  of  O3  =  150°,  hence  £  =  210°. 

For  the  third  harmonic. — The  successive  distances  for  the 
ordinates  to  be  taken  from  the  strip  are,  first  (0  to  2)  +  (4  to  6) 
+(8  to  10),  then  (1  to  3)  +  (5  to  7)  +  (9  to  11).  The  curve 
almost  coincides  with  the  x  axis,  and  as  the  distance  to  the 
crests  has  to  be  divided  by  6  to  obtain  the  amplitude  a'", 


210*  \ 

R--25J 


\ 


Fig.  265 

it  is  evident  that  this  third  term  of  the  harmonic  series  is 
negligible.  Hence  the  equation  may  be  written: 

y  =  3-5  sin(*  +  28-2°)  +  -25  sin(2^  +  210°). 

Now  draw  the  two  harmonics  to  scale  (fig.  265),  recompose, 
compare  the  result  with  the  original  graph,  and  thus  check 
the  work. 


WAVE  MOTION 


469 


Boys  are  always  keen  to  know  what  is  behind  such  un- 
usual procedure.  The  explanation  is  really  very  simple. 

Let  (i),  fig.  266,  be  the  first  harmonic  (the  fundamental), 
(ii)  the  second  harmonic,  (iii)  the  third,  and  (iv)  the  fourth. 
We  can  cut  the  whole  wave  in  (ii)  into  two  equal  and  similar 
parts,  and  slide  the  right-hand  half  along  the  axis  and  superpose 
it  on  the  left-hand  half.  We  may  cut  (iii)  into  three  equal  and 


Fig.  266 


similar  parts,  slide  the  second  and  third  parts  along  and 
superpose  them  on  the  first.  We  may  cut  (iv)  into  four 
equal  and  similar  parts,  and  again  superpose. 

Now  suppose  we  have  a  compound  curve  of  unknown 
composition.  If  it  consisted  of  the  first  harmonic  only,  it 
would  be  just  a  simple  sine  curve,  like  fig.  (i). 

If  the  second  harmonic  is  present,  fig.  (ii)  represents  that 
component.  To  test  for  its  presence,  cut  the  composite  curve 
0  to  2?r  into  two,  slide  along  and  superpose,  add  the  corre- 
sponding ordinates  of  the  two  parts  thus  superposed  (yo+JVe* 
yl  +  y7,  &c.,  algebraically,  of  course),  take  the  average  of 
each  of  these  sums  by  dividing  by  2,  and  plot  the  curve. 
That  curve  is  the  second  harmonic  together  with  any  of  its 
multiples,  if  any  of  these  are  components;  but  the  curve 


470  CRAFTSMANSHIP  IN  MATHEMATICS 

does  not  contain  any  other  harmonic  than  these  multiples; 
i.e.  the  curve  so  obtained  is, 

y  =  a2  s'm(2x  +  oc2)  +  «4  sin(4#  -f  a4)  -f  &c. 

If  the  third  harmonic  is  present,  fig.  (iii)  represents  the 
component.  To  test  for  its  presence,  cut  the  composite  curve 
0  to  2?7  into  three,  slide  along  and  superpose,  add  the  corre- 
sponding ordinates  of  the  3  parts  thus  superposed,  take  the 
average  of  each  sum  by  dividing  by  3,  and  plot  the  curve. 
The  curve  is  the  third  harmonic  together  with  any  of  its 
multiples,  if  any  of  these  are  components,  but  the  curve 
does  not  contain  any  other  harmonic  than  those  multiples; 
i.e.  the  curve  obtained  is, 

y  =  03  sin(3#  -f  oc3)  +  a6  sin(6#  -f  ae)  -f  &c. 

So  with  harmonics  beyond  the  third.  But  these  are 
rarely  required;  they  affect  the  result  too  slightly. — The 
proofs  of  these  rules  are  very  simple,  and  should  be  given. 

Inasmuch  as  there  is  no  advantage  in  giving  for  analysis 
any  composite  curves  containing  harmonics  beyond  the 
third,  this  graphic  work  need  not  be  carried  further.  But 
the  boys  ought  now  to  return  to  the  example  represented  by 
figs.  263  and  264,  and  penetrate  the  mystery  of  the  paper 
strip:  the  additions  from  the  strip  are  really  the  additions 
of  superposed  ordinates  resulting  from  cutting  up  the  com- 
posite curve,  sliding  to  the  left,  and  superposing.  The 
reversal  of  the  strip  is  readily  seen  to  be  a  simple  device  for 
converting  subtraction  into  addition. 

Teachers  who  think  well  of  this  method  of  Professor  Runge 
may  refer  to  Zeitschrift  fur  Mathematik  und  Physik,  Vol.  48, 
443-56. 


Professor  Nunn's  Plan:    the  Principle 

A   much   more   important   curve- decomposition   method 
may   be   briefly   considered.      The   fundamental    principle 


WAVE  MOTION 


471 


underlying  it  is  the  obvious  fact  that  the  total  area  of  either 
a  complete  sine  curve  or  of  a  complete  cosine  curve  is  zero, 
since  it  is  equally  divided  by  the  x  axis.  Professor  Nunn's 
exposition  (Algebra,  pp.  521-3)  is  particularly  illuminating, 
though  I  have  sometimes  found  Sixth  Form  boys,  who  had 


Fig.  267 

not  had  a  good  training  in  solid  geometry,  puzzled  over  the 
geometrical  figures.  I  append  an  outline  of  the  exposition, 
together  with  a  few  new  "  solid  "  figures. 

On  one  side  of  a  line  AB  of  length  /,  draw  the  semi- 
sine  curve  y  =  a  sin-*, 

choosing  any  value  for 
the  amplitude  HK  (= 
a).  On  the  other  side 
draw  similarly  the  curve  A 

A 

y  =  sin-*,  with  ampli- 
tude KL  (=  unity).  Cut 
the  figure  out  and  fold 
it  about  AB  until  the 
planes  of  the  two  curves  Fig.  268 

are  at  right  angles.  Now 

mould  a  solid,  in  clay,  plasticine,  soap,  or  any  similar  soft 
material,  to  fill  up  the  space  between  the  curves. 

In  practice,  the  best  way  to  do  this  is  first  to  mould  a 
rectangular  prism  /  units  long  with  cross-section  KH  X  KL. 

Then  draw  the  curve  a  sin  ^*  on  the  face  of  the  prism 

FCDE  (i),  and  the  curve  sin-*  on  the  top  of  the  prism 
MFEG  (ii).  Pare  off  horizontally  round  the  curve  as  in  (i), 


r- 


472 


CRAFTSMANSHIP  IN  MATHEMATICS 


and  vertically  round  the  curve  as  in  (ii).    The  result  is  (iii), 
the  solid  we  require,  ALBH;    the  plan  of  the  solid  is  the 


77 


figure  bounded  by  AB  and  the  curve  sin-*,  and  the  elevation 
in  the  figure  bounded  by  AB  and  the  curve  a  siny#. 


It  is  important  to  note  that  any  section  of  the  solid  by  a 
plane  at  right  angles  to  AB  is  a  rectangle  (e.g.  RSTV)  whose 

adjacent  sides  are  a  sin-#  and  sin-#,  x  being  the  distance 

i  if 

of  the  section  from  A.  Note  that  the  two  lengths  may  be 
measured  either  on  the  flat  surfaces  behind  and  below  or 
on  the  curved  surfaces  in  front  and  above.  Unless  the  solid 
is  actually  constructed,  many  boys  will  have  difficulty  in 
seeing  this. 

The  area  of  the  section  =  a  sin  nx/l  X  sin  nx/l 
=  a  sm2Ttx/l 

=  2(1  -cos2nx/l) 


i-:—7* 


WAVE  MOTION  473 

that  is,  the  area  of  any  section  of  the  solid  is  equal  to  the 

algebraic  difference  between  a  constant  area  -  and  a  variable 

2 
a 

area  -cos  27rx/l. 
2 

For  convenience,  each  of  these  areas  may  be  looked  upon 
as  rectangles,  each  of  height  a/2.  Thus  the  base  of  the 
former  would  be  unity,  and  that  of  the  latter 


I 


B 


(H) 


cos  2-ncc/l 

I  ' 


Fig.  270 


The  two  rectangles  may  be  regarded  as  cross-sections 
of  two  new  solids  of  length  AB  (—  /)  and  of  uniform  height 
a/2.  Above  are  their  plans.  Note  the  neat,  though  obvious, 


(i) 


(ii) 


device  for  showing  the  width  of  the  second.  Fig.  271  shows 
perspective  sketches  (for  the  sake  of  clearness,  figs.  270  and 
271  are  drawn  very  considerably  out  of  proportion,  compared 
with  figs.  268  and  269). 


474  CRAFTSMANSHIP  IN  MATHEMATICS 

Let  a  plane  at  right  angles  to  AB  cut  the  solids  at  PiP2> 
corresponding  to  RSTV  in  fig.  269  (iii).  Then  the  section 
RSTV  is  equal  to  the  difference  between  the  sections  P1  and 
P2;  and  so  with  any  other  vertical  section.  At  KL  in  fig.  269 
(iii),  the  difference  is  between  Qx  and  Q2,  but  since  Q2  is 
negative,  the  difference  is  the  arithmetical  sum.  This  is  as 
might  be  expected,  for  the  section  on  HL  is  the  full  section 
of  the  original  rectangle.  In  the  case  of  section  P2,  the  width 
cos  27rx/l  is  positive;  in  the  case  of  Q2,  it  is  negative.  Thus 
the  area  of  the  section  P2  must  be  reckoned  positive  and  that 
of  Q2  negative. 

It  follows  that  the  part  of  the  solid  above  AB  in  fig.  270 
(ii)  must  be  reckoned  positive,  and  that  below  AB  negative. 
Hence  we  must  regard  the  total  volume  of  the  solid  in  fig. 
271  (ii)  as  0.  But  the  volume  of  the  solid  in  fig.  269  (iii)  is 
equal  to  the  difference  of  the  volumes  of  the  two  solids  in 
fig.  271  (i)  and  (ii).  Hence  the  volume  of  the  solid  in  fig.  269 
(iii)  is  equal  to  the  volume  of  the  simple  prism  in  fig.  271  (i). 
The  volume  of  the  solid  in  fig.  269  (iii)  is  therefore  al/2. — This 
result  is  always  a  surprise  to  the  boys,  and  they  are  much 
inclined  to  question  it.  They  should  be  made  to  think  about 
it  carefully  and  to  search  for  the  fallacy  they  suspect.  It 
will  pay  to  make  the  boys  work  out  one  or  two  particular 
cases.  Let  them  bear  in  mind  that  the  volume  of  the  rect- 
angular blocks  in  fig.  269  (see  fig.  268)  is  /  X  a  X  1  =  al. 

On  one  or  two  occasions  I  have  known  Sixth  Form 
boys  cut  out  their  models  so  carefully  that,  when  checked 
by  weighing,  the  results  have  been  surprisingly  accurate. 
To  cut  fig.  269  (iii)  out  of  soap,  and  to  weigh  the  model 
against  the  parings,  may  afford  a  very  convincing  check. 


WAVE  MOTION 


475 


The  Principle  Applied 

Consider  the  following  figure,  one  complete  element 
(O  to  2?r)  of  a  composite  wave.  The  problem  is  to  determine 
the  amplitudes  of  the  various  component  harmonics;  that 
done,  the  harmonics  are  easily  drawn.  Since  the  right-hand 
half  of  the  curve  is  the  "  image  "  of  the  left-hand  half,  it  is 
sufficient  to  consider  the  left-hand  half  alone;  call  its  length 
/.  We  will  assume  that  there  are  two  components,  viz.  y  =  av 
simrx/l  and  y  =  a2  smS^x/l,  in  other  words  that  the  given 
curve  is  made  up  of  the  first  and  second  harmonics.  (We 
know  from  the  kind  of  symmetry  that  the  third  harmonic  is 
not  a  component  (see  p.  463).) 

On  the  line  IVTN'  (=  MN  =  /),  draw  the  curve  y  = 
sin  irx/l  inverted,  i.e.  a  sine  curve  with  amplitude  unity;  and 


Fig.  272 


make  a  model  of  the  solid  determined  by  the  two  curves 
when  the  lines  M'N'  and  MN  are  made  to  coincide  and 
the  planes  of  the  figures  are  at  right  angles.  Note  that  any 
section  FGK  at  right  angles  to  MN  is  rectangular,  as  in  the 
solid  of  fig.  269.  The  solid  is  not  an  easy  one  to  model 
accurately  (fig.  273). 

The   volume  of   the    composite    solid    is    equal    to   the 


476  CRAFTSMANSHIP  IN  MATHEMATICS 

sum  of  the  two  solids  determined  by  the  curves, 

(i)  y  =  smnx/l  and  y  =  a^  smnx/l, 
and  (ii)  y  =  sin  roe//  and  y  =  az  s'm2nx/l. 

But  the  latter  of  these  volumes  is  easily  proved  equal  to  0 
(cf.  fig.  271  (ii)),  and  the  volume  of  the  former  is  ax//2  (cf. 
fig.  271  (i)).  Hence 

total  volume  of  solid  =  aJ/2 (i) 


Fig.  273 

But  the  volume  may  also  be  determined  directly,  by 
calculating  the  mean  value  of  its  cross-section.  Consider,  for 
instance,  the  vertical  section  at  G  on  MN  where  MG  =  2//3 
=  x,  so  that  TTX/I  =  277/3  radians  or  120°.  The  section  in  a 
rectangle  whose  sides  FG,  GK  are  closely  analogous  to  the 
sides  RS,  ST  in  fig.  269  (iii).  Of  these  two  sides,  FG  may 
easily  be  determined  by  actual  measurement  from  the  curve, 
while  GK  =  sin!20°  =  \/3/2-  The  product  gives  the  area 
of  the  vertical  section  through  FGK. 

In  this  way  we  may  find  the  area  of  any  number  of  such 
vertical  sections.  For  convenience,  divide  MN  into  12  equal 
parts,  Calculate  the  areas  of  the  respective  sections  through 


WAVE  MOTION  477 

the  dividing  points,  and  then  by  Simpson's  rule*  the  volume 
of  the  solid.  Deduce  from  this  the  average  cross -section 
Ax  by  dividing  by  /. 

Thus         vol.  =  A!/. 
But         vol.  =  aJ/2         (by  (i)). 

"ll  -  A  / 
T~  Al/* 

or  flj  =  2Aj. 

In  a  similar  manner,  by  supposing  a  second  solid  to  be 
formed  by  combining  the  given  half  curve  with  the  curve 
y  =  sin2?rjc//,  the  value  of  a2  may  be  determined.  If  the 
given  curve  contained  any  other  harmonic  components, 
their  amplitudes  might  be  determined  in  the  same  way. 

The  principle  of  the  method  is  that  any  sine  curve  y  = 
sin  rnx/l  when  combined  with  half  the  given  composite  curve 
determines  a  solid  whose  volume  (arl/2)  depends  on  the 
amplitude  ar  of  the  component  y  =  ar  sin  mx/l,  and  not  at 
all  on  the  amplitude  of  any  other  component.  In  this  way, 
the  successive  sine  components  can  be  dealt  with  one  by 
one,  and  their  amplitudes  determined.  The  determination 
of  the  amplitudes  is,  of  course,  the  very  essence  of  the  problem. 

The  work  of  computing  the  average  cross-sections  can  be 
divided  up  amongst  the  members  of  the  class.  Instruct  them 
to  carry  out  the  following  operations,  and  to  tabulate  the 
results: 

(1)  To  determine  the  amplitude  of  the  first  harmonic. 

(a)  Divide  up  MN  into  twelve  15°-phase  differences; 
erect  the  ordinates  and  measure  their  lengths  in  millimetres. 
In  accordance  with  Simpson's  rule,  only  half  the  height 
of  the  first  and  last  ordinates  is  required  in  the  calculations, 
but  as,  in  this  instance,  these  happen  to  be  zero,  the  halving 
makes  no  difference. 

(/J)  Calculate  the  successive  values  of  sinn!5°,  n  being  the 
number  of  the  ordinates. 

*  "  Add    half  the  first  and  last  areas  and  the  whole  of  the  intermediate  areas, 
and  multiply  the  sum  by  the  common  interval." 


478 


CRAFTSMANSHIP  IN  MATHEMATICS 


(y)  Multiply  (a)  by  (/?)  and  so  obtain  the  areas  of  the 
successive  sections  of  the  solid  (fig.  273). 


i. 

ii. 

1st  Harmonic. 

2nd  Harmonic. 

Number 
of 
Ordinate. 

Length  of 
Ordinate 
in  mm. 

iii. 

iv. 

V. 

vi. 

vii. 

viii. 

Angle. 

Sine. 

Area 
ii  or  iv. 

Angle. 

Sine. 

Area 
ii  or  iv. 

0 

0 

0° 

0 

0 

0° 

0 

+o 

1 

13-0 

15° 

•26 

3-38 

30° 

•5 

+  6-50 

2 

23-0 

30° 

•5 

11-50 

60° 

•87 

+  19-93 

3 

29-5 

45° 

•71 

20-90 

90° 

1-0 

4-29-50 

4 

31-5 

60° 

•87 

27-40 

120° 

•87 

4-27-40 

5 

29-5 

75° 

•97 

28-60 

150° 

•5 

4-14-75 

6 

24-0 

90° 

1-0 

24-00 

180° 

0 

0 

7 

17-5 

105° 

•97 

16-90 

210° 

—  •5 

-8-75 

8 

10-3 

120° 

•87 

8-92 

240° 

-•87 

-8-92 

9 

5-0 

135° 

•71 

3-54 

270° 

-1-0 

-5-00 

10 

2-0 

150° 

•5 

1-00 

300° 

-•87 

-1-74 

11 

1-0 

165° 

•26 

•26 

330° 

—  -5 

-•50 

12 

0 

180° 

•0 

0 

360° 

0 

0 

Total  = 
Average  area  —  Ax  = 

146-40 

Total  = 

73-16 

12-20 

Av'ge  =  A2  = 

6-097 

Amplitude  of  the  1st  harmonic  = 


fll  =  2Aj  =  (12-2  mm.  X  2) 
=  24-4  mm. 
=  2-44  cm. 


(2)  To  determine  the  amplitude  of  the  second  harmonic.  — 
Corresponding  to  the  half  curve  MN  will  be  a  complete 
sine  curve  of  the  second  harmonic.  Hence  the  angles  will 
now  be  #30°,  and  the  sines  from  180°  to  360°  will  be  negative 
The  ordinate  lengths  will  be  the  same  as  before. 


Amplitude  =  aa  = 


=  (6-097  mm.  X  2) 
=  1-22  cm. 


WAVE  MOTION  479 

Hence  the  original  curve  is, 

y  =  2-44  sinnx/l+  1-22  sin2nx/l. 

The  periods  being  known,  and  the  amplitudes  having  been 
found,  the  angles  follow  at  once. 

Let  the  boys  realize  fully  that  the  essence  of  the  problem 
is  the  discovery  of  the  amplitudes  of  the  component  harmonics. 


31-5 


274 


By  improvising  the  solids  and  devising  two  different  schemes 
for  determining  their  volumes,  we  obtain  two  different  formulae 
each  involving  a  in  terms  of  A.  It  is  true  that  A  appears  as 
an  area,  but,  by  taking  one  of  the  dimensions  of  the  area  as 
unity,  A  becomes  a  linear  value,  and  of  course  we  begin  by 
giving  the  solid  a  base  consisting  of  a  sine  curve  of  unit 
amplitude. 

The  subject  can  be  followed  up  by  integration.  The  boys 
are  now  ready  for  it,  for  they  have  learnt  what  the  subject 
is  really  about. 

Books  to  consult: 

1.  The  Teaching  of  Algebra,  Nunn. 

2.  Manual  of  Practical  Mathematics,  Castle. 

3.  Any  modern  standard  work  on  Sound. 


480  CRAFTSMANSHIP  IN  MATHEMATICS, 

CHAPTER  XXXII 

Mechanics 

The  Teacher  of  Mechanics 

The  most  successful  teachers  of  mechanics  whom  I  have 
known  are  those  who  have  had  a  serious  training  in  a  me- 
chanical laboratory;  who  know  something  of  engineering,  and 
are  familiar  with  modern  mechanism;  who  are  competent 
mathematicians;  and  who  have  mastered  Mach's  Mechanics, 
especially  Chapters  I  and  II.*  Mach's  book  is  universally 
recognized  as  the  book  for  all  teachers  of  mechanics.  It 
deals  with  the  development  of  the  fundamental  principles 
of  the  subject,  traces  them  to  their  origin,  and  deals  with 
them  historically  and  critically.  The  treatment  is  masterly. 
The  book  might  with  advantage  be  supplemented  by  Stallo's 
Concepts  of  Modern  Physics  (now  out  of  date  from  some 
points  of  view),  Karl  Pearson's  Grammar  of  Science,  and 
Clifford's  Common  Sense  of  the  Exact  Sciences  and  Lectures 
and  Essays  (still  first-rate,  though  written  50  years  ago). 

It  is  of  great  advantage  to  a  teacher  of  mechanics  to  be 
familiar  with  the  subject  historically.  The  main  ideas  of  the 
subject  have  almost  always  emerged  from  the  investigation  of 
very  simple  mechanical  processes,  and  an  analysis  of  the 
history  of  the  discussions  concerning  these  is  the  most 
effective  method  of  getting  down  to  bedrock. 

Who  were  the  great  investigators?  The  scientific  treat- 
ment of  statics  was  initiated  by  Archimedes  (287-212  B.C.), 
who  is  truly  the  father  of  that  branch  of  mechanics.  The 
work  he  did  was  amazing,  but  there  was  then  a  halt  for  1700 
or  1800  years,  when  we  come  to  Leonardo,  Galileo,  Stevinus, 
and  Huygens;  to  Torricelli  and  Pascal;  and  to  Guericke 
and  Boyle.  For  dynamics,  we  go  first  to  its  founder  Galileo 

*  Hertz  also  wrote  a  Mechanics  of  the  same  masterly  kind,  but  there  is  no  English 
translation,  so  far  as  I  know. 


MECHANICS  481 

(falling  bodies,  and  motion  of  projectiles),  then  to  Huygens 
(the  pendulum,  centripetal  acceleration,  magnitude  of  acceler- 
ation due  to  gravity),  and  then  to  Newton  (gravitation,  laws 
of  motion).  The  great  principles  established  by  Newton 
have  been  universally  accepted  almost  down  to  the  present 
time,  and,  so  far  as  ordinary  school  work  is  concerned,  will 
continue  to  be  used — at  least  during  the  present  generation. 

A  boy  is  always  impressed  by  Newton's  argument  that 
since  the  attraction  of  gravity  is  observed  to  prevail  not 
only  on  the  surface  of  the  earth  but  also  on  high  mountains 
and  in  deep  mines,  the  question  naturally  arises  whether  it 
must  not  also  operate  at  greater  heights  and  depths,  whether 
even  the  moon  must  not  be  subject  to  it.  And  the  boy  is 
still  more  impressed  by  the  story  of  the  success  of  Newton's 
subsequent  investigation. 

Newton's  four  rules  for  the  conduct  of  scientific  investi- 
gation (regulce  philosophandi)  are  the  key  to  the  whole  of  his 
work,  and  should  be  borne  in  mind  by  his  readers. 

The  First  Stage  in  the  Teaching  of  Mechanics 

How  do  successful  teachers  begin  mechanics  with  boys  of 
about  12  or  13?  They  usually  begin  by  drawing  upon  the 
boys'  stock  of  knowledge  of  mechanism.*  Most  boys  know 
something  of  mechanism,  some  will  have  had  enough  curiosity 
to  discover  a  great  deal,  and  a  few  will  probably  have  had 
experience  of  taking  to  pieces  machines  of  some  sort  and  of 
putting  them  together  again.  This  stock  of  knowledge  may  be 
sorted  out,  and  the  topics  classified  and  made  the  subjects 
of  a  series  of  lessons.  By  means  of  an  informal  lesson  on  some 
piece  of  mechanism,  an  important  principle  may  often  be 
worked  out,  at  least  in  a  rough  way. 

I  have  known  a  teacher  give  his  first  lesson  on  mechanics 
in  the  school  workshop,  utilizing  the  power-driven  lathe  and 
the  drilling-machine;  another  first  lesson  in  the  school  play- 
ground, an  ordinary  bicycle  being  taken  to  pieces.  I  have 

•See  Chapter  VIII,  Science  Teaching. 
(£291)  32 


482  CRAFTSMANSHIP  IN  MATHEMATICS 

seen  a  model  steam-engine  used  for  the  same  purpose, 
and  I  have  known  beginners  taken  to  a  local  farm  to  watch 
agricultural  machinery  at  work.  In  all  these  instances  the 
boys  learnt  that  their  new  subject  seemed  to  have  a  very 
close  relation  with  practical  life.  They  were  not  made  to 
look  upon  it  as  another  branch  of  mathematics,  and  a  rather 
difficult  branch  at  that. 

Let  the  early  lessons  be  lessons  to  establish  very  simple 
principles.  Never  mind  refinements  and  very  accurate 
measurements.  Do  not  bother  about  small  details,  and  avoid 
all  complications.  Let  the  boy  get  the  idea,  and  get  it  clearly. 
Very  simple  arithmetical  verifications  are  quite  enough  at 
this  stage.  The  boy's  curiosity  is  at  first  qualitative;  let 
that  be  whetted  first,  and  then  turned  into  a  quantitative 
direction  gradually.  Encourage  the  boy  to  find  out  things  for 
himself,  and  do  not  tell  him  more  than  is  really  necessary. 
Encourage  him  to  ask  questions,  but  as  often  as  possible 
answer  these  by  asking  other  questions  which  will  put  him 
on  a  new  line  of  inquiry.  Let  him  accumulate  knowledge 
of  machines  and  machine  processes.  Give  him  some  scales 
and  weights,  and  a  steelyard,  and  tell  him  just  enough  to 
enable  him  to  discover  the  principle  of  moments,  but  do 
not  talk  at  first  about  either  "  principle  "  or  "  moments  ". 
It  is  good  enough  if  at  this  stage  he  suggests  that 

long  arm  X  little  weight  — -  short  arm  X  big  weight. 

He  has  the  idea,  and  the  idea  is  expressed  in  such  a  form  that 
it  sticks.  Give  him  a  model  wheel  and  axle,  give  him  a  hint 
that  it  is  really  the  lever  and  the  lever-law  over  again,  and 
make  him  show  this  clearly.  Give  him  some  pulleys  and  let 
him  discover,  with  the  help  of  one  or  two  leading  questions, 
how  a  small  weight  may  be  made  to  pull  up  a  big  weight, 
and  let  him  work  out  the  same  law  once  more,  but  now  in  the 
form  that  what  is  gained  in  power  is  lost  in  speed.  Give  him 
a  triangular  block  and  an  endless  chain,  let  him  repeat  Ste- 
vinus'  experiment,  and  so  discover  the  secret  of  the  inclined 
plane.  Let  him  use  a  jack  to  raise  your  motor-car  (and  inci- 


MECHANICS  483 

dentally  learn  something  about  "  work  ");  now  tell  him  some- 
thing about  the  pitch  of  the  screw,  something  about  Whit- 
worth's  device  for  measuring  very  small  increases  in  length, 
something  about  the  manufacture  of  a  Rowlands  grating. 
Encourage  him  to  give  explanations  of  mechanical  happenings 
in  everyday  life,  and  use  his  suggestions  as  pegs  on  which  to 
hang  something  new. 

A  term  of  this  kind  of  work  pays.  The  boy  is  accumulating 
knowledge  of  the  right  sort,  and  when  the  subject  is  taken  up 
more  formally  and  with  a  more  logical  sequence,  rapid  progress 
may  be  made.  Once  he  has  been  taught  to  read  elementary 
mechanism,  it  is  easy  enough  to  teach  him  its  grammar. 
Surely  this  is  the  right  sequence.  Mechanism  must  come 
before  mechanics.  The  mathematics  of  the  subject  is  a  super- 
structure, to  be  built  upon  a  foundation  of  clear  ideas. 

Of  course,  if  the  preliminary  work  of  the  preparatory  school 
or  department  has  been  properly  done,  the  way  is  paved  for 
an  earlier  treatment  of  a  more  formal  kind. 

The  Second  Stage 

The  second  stage  should  consist  of  work  of  a  more  syste- 
matic character,  but  still  work  essentially  practical,  though 
arranged  on  a  logical  string.  Ideas  will  now  be  classified,  and 
mathematical  relations  gradually  introduced.  But  the  physical 
thing  and  the  physical  action  must  still  remain  in  the  front  of 
the  boy's  mind.  The  mathematics  will  take  care  of  itself. 

Let  the  teaching  be  inductive  as  far  as  possible.  Obtain 
all  .necessary  facts  from  experiments,  and  do  not  use  experi- 
ments merely  for  verifying  a  principle  enunciated  dogmatically. 

The  basic  principles  to  be  taught  are  really  very  few,  and 
a  boy  who  knows  these  thoroughly  well  can  work  most  ordinary 
problems  on  them.  Mechanics  is,  after  all,  largely  a  matter  of 
common  sense.  The  laws  of  equilibrium,  together  with  the 
ratio  of  stress  to  strain,  covers  almost  the  whole  range  of 
statical  problems,  including  those  of  hydrostatics;  while  New- 
ton's Laws  of  Motion  covers  practically  everything  else.  But 


484  CRAFTSMANSHIP  IN  MATHEMATICS 

of  course  these  are  basic  principles.  If  they  are  known, 
known,  derived  principles  are  learnt  easily  enough;  if  they  are 
only  vaguely  known,  derived  principles  are  never  really 
mastered. 

Statics  or  dynamics  *  first?  Teachers  do  not  agree.  There 
is  much  to  be  said  for  beginning  with  dynamics,  first  using 
the  ballistic  balance  for  studying  colliding  bodies,  and  the 
momentum  lost  by  one  and  gained  by  another;  it  is  then  an 
easy  step  to  pass  on  to  the  idea  of  force.  But  a  boy  who  is 
led  to  think  of  a  force  as  something  analogous  to  muscular 
effort  will  always  be  in  trouble,  and  in  any  case  he  is  likely 
to  form  a  very  vague  idea  of  acceleration.  Of  course,  uniform 
acceleration  is  anything  but  common  in  practical  life:  we 
nearly  always  refer  either  to  falling  bodies  or  to  a  train 
moving  out  from  a  station.  And  it  is  this  difficulty  that 
makes  many  teachers  take  up  statics  first.  Although,  at  the 
outset,  a  boy's  working  idea  of  force  is  necessarily  crude, 
a  spring  balance,  for  simple  quantitative  experiments,  helps 
to  put  the  boy  on  the  right  track,  and  there  is  much  to  be 
said  for  allowing  him  to  assume,  to  begin  with,  that  weight 
is  the  fundamental  thing  to  be  associated  with  force.  At 
an  early  stage  he  may  verify,  to  his  own  satisfaction,  the 
principles  of  the  parallelogram  and  triangle  of  forces,  but  he 
must  be  warned  that  he  has  not  yet  "  proved  "  these  principles 
and  cannot  yet  do  so.  But  since  the  parallelogram  of  forces  is 
such  a  useful  working  principle,  it  would  be  foolish  not  to 
allow  the  boy  to  use  it  before  he  can  prove  it  formally.  At 
this  stage  formal  proofs  are  difficult,  and  it  is  simply  dis- 
honest to  encourage  a  boy  to  reproduce  a  page  of  bookwork 
giving  a  proof  of  something  quite  beyond  his  comprehension, 
though  this  was  common  enough  thirty  or  forty  years  ago. 

Do  not  employ  graphic  statics  at  too  early  a  stage,  or  the 
real  point  at  issue  may  be  obscured. 

Now  as  to  dynamics.  What  is  the  best  approach?  We 
have  already  referred  to  the  ballistic  balance.  Should  At- 
wood's  machine  be  used?  It  may  be  used,  perhaps,  for 

*  The  terms  kinetics  and  kinematics  are  falling  into  disuse. 


MECHANICS  485 

illustrating  the  laws  of  motion,  but  not  as  a  practical  method 
of  finding  g. 

Atwood's  machine  has  been  superseded  by  Mr.  Fletcher's 
trolley,*  by  means  of  which  practically  the  whole  of  the  prin- 
ciples of  dynamics  may  be  satisfactorily  demonstrated.  It 
lends  itself  to  many  experiments,  all  of  which  provide  a 
space-time  curve  ready  made,  and,  from  that,  speed-time 
and  acceleration-time  curves  may  be  plotted.  In  a  paper 
read  at  the  York  meeting  of  the  British  Association,  Mr.  C.  E. 
Ashford  gave  details  of  a  large  number  of  trolley  experiments 
as  performed  at  Dartmouth,  a  school  where  the  teaching 
of  mechanics  is  well  known  to  be  of  a  high  order.  Reference 
should  be  made  to  Mr.  Fletcher's  own  article  in  the  School 
World  for  May,  1904.  In  it  he  shows  how  boys  may  be 
given  sound  ideas  of  the  physical  meaning  of  the  terms, 
moment  of  inertia,  angular  momentum,  moment  of  momen- 
tum, and  therefore  of  moment  of  rate  of  change  of  momentum 
and  moment  of  force.  Useful  teaching  hints  may  also  be  found 
in  Mr.  S.  H.  Wells's  Practical  Mechanics  and  Mr.  W.  D. 
Eggar's  Mechanics. 

Once  the  foundations  of  mechanics  have  been  well  and 
truly  laid  the  superstructure  may  be  erected  according  to 
traditional  methods.  To  leave  the  subject  just  as  developed 
in  the  laboratory  would  be  to  leave  it  unfinished.  But  the 
superstructure  may  now  be  built  properly.  When  necessary 
formulae  have  been  evolved  from  experiment,  the  physical 
things  behind  the  formulae  have  to  the  boy  a  reality  of  mean- 
ing which  the  older  "  methods  of  applied  mathematics  " 
teaching  could  not  possibly  give  him. 

If  principles  are  not  understood,  proofs  have  no  meaning. 

Throughout  the  whole  of  a  mechanics  course  every  oppor- 
tunity should  be  taken  to  excite  the  boys'  interest  in  new 

mechanical  inventions.     It  helps  the  more  academic  work 

• 

*The  friction  of  the  trolley  may  be  eliminated  either  by  tilting  the  plane  to  the 
necessary  angle,  or  by  attaching  a  weight  that  will  just  maintain  uniform  motion. 
The  friction  of  the  pulley  over  which  the  thread  passes  cannot  be  compensated,  and 
it  is  therefore  necessary  to  use  a  good  pulley. 


486  CRAFTSMANSHIP  IN  MATHEMATICS 

enormously,  and  makes  the  boys  feel  that  the  subject  is  really 
worth  taking  trouble  over.  Examples  occur  on  every  side — 
variable  speed  gears,  transmission  gears,  taximeters,  boat- 
lowering  gear,  automatic  railway  signalling,  automatic  tele- 
phones, the  self-starter  in  a  motor-car,  the  kick-starter  in  a 
motor-cycle,  and  so  on.  Some  mechanical  devices  depend,  in 
their  turn,  on  electricity,  and  their  place  of  introduction  into 
a  teaching  course  would  be  determined  accordingly.  Complex 
mechanisms  like  the  air-plane,  the  submarine,  the  paravane, 
should  not  be  wholly  forgotten.  Boys  can  read  up  such  things 
for  themselves,  and  perhaps  prepare  and  read  papers  on  them 
to  the  school  science  society. 

Hydrostatics 

The  mechanics  of  fluids  is  an  exceedingly  difficult  subject 
to  teach  effectively.  Even  a  Sixth  Form  boy  is  sometimes  held 
up  by  questions  on  the  barometer  or  on  Dulong  and  Petit's 
equilibrating  columns.  The  work  of  Archimedes  and  Pascal 
for  liquids  and  of  Boyle  for  gases  cannot  be  too  well  done. 
Above  all,  the  U-tube  must  receive  careful  attention,  and 
especially  the  surface  level  above  which  pressures  are  compared. 
Do  not  buy  Hare's  apparatus  from  an  instrument-maker's. 
The  standard  pattern  is  always  made  with  two  straight  tubes, 
of  the  same  bore,  fixed  vertically.  Let  the  boys  make  a  variety 
of  forms  of  this  apparatus  for  themselves,  and  work  out  the 
vertical  height  law  from  data  as  varied  as  possible.  Approach 
the  whole  subject  of  hydrostatics  from  the  point  of  view  of 
familiar  phenomena,  e.g.  measure  the  water  pressure  from  a 
tap  in  the  basement  and  again  from  a  tap  in  the  top  story  of 
the  school,  and  see  if  there  is  any  sort  of  relation  between  the 
difference  of  these  pressures  and  the  height  of  the  school. 
Do  not  try  to  establish  a  principle  formally  until  the  phenomenon 
under  investigation  is  clearly  understood  as  a  physical  happening. 
Let  boys  know  really  what  they  are  going  to  measure  before 
they  begin  to  measure.* 

•The  preceding  paragraphs  are  taken  from  Science  Teaching  (pp.  121-8). 


MECHANICS  487 

The  Johannesburg  British  Association  Meeting 

At  the  Johannesburg  meeting  of  the  British  Association, 
an  animated  discussion  took  place  on  the  general  question 
of  the  teaching  of  mechanics.  It  followed  on  a  paper  read  by 
Professor  Perry.  We  append  a  few  suggestive  extracts. 

Professor  Perry. — "  The  very  mathematical  man  often 
does  not  know  anything  of  mechanics;  it  is  the  subject  of 
applied  mathematics  that  he  has  studied  and  that  he  cares 
for. 

"  The  two  elementary  principles  of  statics,  (1)  if  forces 
are  in  equilibrium,  their  vector  sum  is  zero,  and  (2)  the  sum 
of  their  moments  about  any  axis  whatsoever  is  zero,  ought 
to  be  so  clear  to  a  pupil  that  it  is  practically  impossible  for 
him  to  forget  them.  They  ought  to  be  as  much  a  part  of  his 
mental  machinery  as  the  power  to  walk  is  part  of  his  physical 
function. 

"  I  lay  no  stress  upon  mere  abstract  proofs  of  propositions 
in  mechanics.  When  understanding  is  affected  there  is  no 
difficulty  about  the  proofs.  It  is  quite  usual  to  find  men  who 
can  prove  everything,  without  having  any  comprehension  of 
what  they  have  proved/' 

Mr.  W.  H.  Macaulay. — "  I  agree  with  the  taking  of  statics 
before  dynamics.  I  also  agree  that  graphical  statics  is  a  subject 
full  of  dodges,  though  very  good  to  learn  if  you  want  to  use 
them  every  day." 

Professor  Boys. — "  I  absolutely  agree  as  to  the  desirability 
of  dealing  with  fundamental  principles,  and  of  not  worrying 
about  innumerable  details.  ...  A  friend  of  mine  heard 
Lord  Kelvin  say  in  one  of  his  lectures,  '  And  now  we  come 
to  the  principle  of  the  lever.  You  will  understand  that  levers 
are  divided  into  three  orders,  levers  of  the  first  order,  of  the 
second,  and  of  the  third — but  which  of  them  is  which  I 
cannot  for  the  life  of  me  tell  you/  Textbooks  were  at  one 
time  filled  up  with  futile  and  unnecessary  kinds  of  dis- 
crimination which  had  nothing  whatever  to  do  with  the  sub- 
ject." 


488  CRAFTSMANSHIP  IN  MATHEMATICS 

Professor  Bryan. — "  The  idea  of  mechanics  which  appeals 
most  readily  to  a  young  boy  is  that  it  has  something  to  do 
with  machines,  and  that  machines  have  something  to  do  with 
turning  out  useful  work.  There  is  no  better  way  of  stimulating 
interest  in  the  subject  than  showing  the  beginner  that  when 
you  have  got  your  machine  for  changing  one  kind  of  work 
into  another,  you  are  no  better  off  than  when  you  started. " 

Professor  Hicks. — "  My  own  experience  is  in  approaching 
mechanics  from  a  kinetic  point  of  view.  First  let  the  boys 
find  out  by  experiment  that  momentum  remains  constant. 
Of  course  the  first  thing  depends  on  what  mass  is;  then  we 
must  proceed  to  show  that  when  two  bodies  collide  with  equal 
velocities  they  come  to  rest.  By  making  experiments  of 
velocities  of  colliding  bodies,  boys  get  to  realize  that  momentum 
remains  unalterable.  Given  two  colliding  bodies  in  a  straight 
line,  the  momentum  lost  by  one  is  gained  by  the  other. 
By  getting  a  large  number  of  experiments,  pupils  come  to 
a  realized  knowledge  of  that." 

Sir  David  Gill. — "  I  remember  Clerk  Maxwell  illustrating 
the  misuse  of  definitions  by  a  funny  story.  He  said  he  went 
into  his  room  one  day,  and  there  was  a  white  cat  which 
jumped  out  of  the  window.  He  and  his  friends  ran  to  the 
window  to  see  what  had  become  of  the  cat,  and  the  animal 
had  disappeared,  no  one  being  able  to  solve  the  mystery. 
At  last  he  solved  the  problem.  He  said  it  must  be  this.  The 
white  cat  jumped  out  of  the  window,  fell  a  certain  distance 
with  a  certain  velocity,  and  collided  with  an  ascending  black 
cat.  There  were  therefore  two  equal  and  opposite  cats 
meeting  with  equal  and  opposite  velocities,  the  result  being 
no  cat. — Without  a  proper  understanding  of  definitions  of 
these  things,  one  might  arrive  at  such  an  absurdity  as  this 
story  illustrates." 

Professor  Forsyth. — "  The  first  stage  in  teaching  mechanics 
is  not  the  stage  in  which  pupils  have  to  prove,  or  attempt  to 
prove,  or  can  be  expected  to  prove,  anything.  That  belongs 
to  a  later  stage.  The  first  thing  to  do  is  accustom  the  pupils 
to  the  ordinary  relations  of  bodies  and  of  their  properties." 


MECHANICS  489 

Mr.  W.  D.  Eggar. — "  I  should  like  to  see  a  penny-in- 
t he-slot  automatic  weighing  machine  in  every  passenger  lift, 
so  that  the  fundamental  experiment  of  showing  a  connexion 
between  force  and  acceleration  could  be  within  the  reach  of 
everybody." 

Professor  Minchin. — "  I  hope  to  see  the  term  *  centrifugal 
force  '  utterly  banished." 

Mr.  C.  Godfrey. — "  Statics  is  a  fairly  easy  matter  if  one 
begins  with  experiment.  Nor  need  experiment  cease  after 
the  first  stage;  any  school  should  be  able  to  get  hold  of  some 
bit  of  machinery  with  plenty  of  friction  in  it,  say  a  screw- 
jack,  and  investigate  efficiency.  Plotting  '  load  '  against 
*  effort  '  leads  to  very  striking  results. 

"  There  is  the  question  of  mass  and  weight.  In  vain  one 
resorts  to  the  centre  of  the  earth;  it  is  all  too  hypothetical. 
I  remember  as  a  boy  being  puzzled  to  understand  how  the 
weight  of  a  train  (acting  vertically)  could  have  anything  to 
do  with  its  acceleration  under  a  pull  (horizontal)  from  the 
engine. 

"  We  might  give  a  touch  of  reality  to  the  kinetics  course 
by  brake  horse-power  determinations.  It  should  be  possible 
to  rig  up  for  a  few  shillings  a  brake-drum  on  a  motor  (electric 
or  water);  even  a  motor-cycle  on  a  stand  or  a  foot-lathe 
might  serve  the  purpose. 

"  Engineers  talk  in  a  very  confusing  way  about  centrifugal 
force.  When  a  particle  moves  in  a  circle  uniformly,  the  force 
on  the  particle  is  centripetal  and  the  force  on  the  constraints 
is  centrifugal.  But  the  popular  use  of  language  and  the 
popular  belief  is  that  there  is  an  outward  force  on  the  particle." 

"  Applied  "  Mathematics 

The  old  school  of  "  pure  "  mathematicians  very  cleverly 
picked  out  from  the  whole  subject  of  mechanics  and  engineer- 
ing such  problems  as  lent  themselves  to  algebraic  and 
geometrical  treatment,  and  left  the  residue,  rather  disdain- 
fully labelled  "  applied  mechanics  ",  to  be  dealt  with  by 


490  CRAFTSMANSHIP  IN  MATHEMATICS 

teachers  of  lower  degree.  Note  the  term  "  applied  ".  The 
real  mechanics  was  the  mechanics  that  could  be  done  from 
an  easy  chair,  and  was  a  mathematicians'  job.  The  building 
of  the  Assouan  dam  and  of  the  Forth  Bridge  were  trivial 
things  which  any  "  ordinary  engineer  "  could  take  in  hand, 
trivial  things  that  had  no  relation  whatever  to  "  pure " 
thought.  This  temper  survived  even  until  the  present  century. 
When  the  two  Wrights  were  risking  their  lives  by  experiment- 
ing with  the  first  air-plane,  a  well-known  mathematician  wrote 
to  the  press  protesting  against  such  folly,  inasmuch  as  mathe- 
maticians had  not  yet  worked  out  the  mathematical  principles 
of  flight! 

The  mathematician's  proper  share  of  such  work  is  to 
begin  where  the  inventor  or  the  engineer  leaves  off;  it  is 
not  his  business  to  invent  paper  air-planes,  but  to  learn 
from  the  real  thing  the  principles  of  flight  and  to  see  if  these 
rest  on  secure  mathematical  foundations;  if  they  do  not, 
he  may  be  able  to  offer  fruitful  suggestions.  Of  course  if 
the  mathematician  happens  to  have  been  trained  as  an 
engineer,  that  is  a  different  matter.  Unless  the  teacher  of 
mechanics  knows  something  of  actual  engineering,  his  me- 
chanics is  likely  to  have  but  a  remote  connexion  with  actual 
mechanism.  There  are  still  teachers  of  mechanics  who  have 
had  neither  workshop  nor  laboratory  experience,  and  naturally 
they  tend  to  shirk  those  parts  of  the  subject  that  do  not 
come  within  the  four  corners  of  algebra  and  geometry.  It 
is  not  an  uncommon  thing  for  a  course  of  lessons  on  ele- 
mentary statics  to  include  not  a  single  word  about,  for  instance, 
the  equilibrium  and  stability  of  walls,  the  effect  of  buttresses, 
the  thrust  along  rafters,  or  about  roof-trusses  or  cranes. 
Friction  may  be  the  subject  of  a  lesson  with  no  mention 
whatever  of  lubricants.  Energy  may  be  the  subject  of  others, 
and  yet  no  reference  be  made  to  energy  storage  in,  for  example, 
accumulators  and  fly-wheels.  The  transmission  of  motion 
and  power  is  rarely  touched  upon  seriously  in  a  course  of 
mechanics  lessons.  And  yet  all  such  things  as  are  thus  ignored 
are  just  those  things  that  have  already  been  included,  in 


MECHANICS  491 

some  measure,  within  the  four  corners  of  the  boys'  daily 
experience.  Subjects  like  tension  and  compression,  shearing 
and  torsion,  beams,  girders,  and  frameworks,  are  passed 
over  hurriedly  as  of  little  importance.  Why  is  elementary 
hydrostatics  so  often  given  such  short  shrift?  Why  is  it  not 
followed  up  by  the  subject  which  really  matters,  viz.  ele- 
mentary hydraulics — the  flow  of  water  through  orifices  and 
pipes,  the  pressure  in  a  water-main,  water-wheels,  turbines, 
the  propulsion  of  ships  and  air-planes,  and  hydraulit  machines? 
As  for  capillarity  and  surface  tension,  which  lend  themselves 
to  all  sorts  of  delightful  experiments,  they  are  too  often  an 
affair  of  just  blackboard  and  chalk.  Do  not  put  off  that 
interesting  section  of  physics,  "  properties  of  matter  "  (the 
twin-sister  of  mechanics),  until  the  Sixth  Form.  The  mathe- 
matics of  it  in  the  Sixth,  yes;  but  the  necessafy  laboratory 
course  can  be  taken  in  the  Fourth  and  Fifth. 

In  short,  the  mathematics  of  mechanics  is  very  serious 
Sixth  Form  work.  The  practical  work  that  must  be  done 
before  the  mathematical  work  can  profitably  be  attempted 
may  be  done  earlier. 

We  will  give  an  extract  from  an  elementary  textbook 
on  Mechanics  for  Beginners,  with  a  very  well  known  name 
on  the  title-page.  It  is  an  introduction  to  Moment  of 
Inertia. 

"  Let  the  mass  of  every  particle  of  a  body  be  multiplied 
into  the  square  of  its  distance  from  an  assigned  straight  line; 
the  sum  of  these  products  is  called  the  moment  of  inertia  of 
the  body  about  that  straight  line.  The  straight  line  is  often 
called  an  axis. 

"  The  moment  of  inertia  of  any  body  about  an  assigned 
axis  is  equal  to  the  moment  of  inertia  of  the  body  about  a 
parallel  axis  through  the  centre  of  gravity  of  the  body,  increased 
by  the  product  of  the  mass  of  the  body  into  the  square  of  the 
distance  between  the  axes. — Let  m  be  the  mass  of  one  particle 
of  the  body;  let  this  particle  be  at  A.  Suppose  a  plane  through 
A,  at  right  angles  to  the  assigned  axis,  to  meet  the  axis  at 


492  CRAFTSMANSHIP  IN  MATHEMATICS 

O,  and  to  meet  the  parallel  axis  through  the  centre  of  gravity 
A     at  G.     From  A  draw  a  straight  line 
AM,  perpendicular  to  OG  or  to  OG 
produced.    Let  GM  =  x,  where  x  is 
a  positive  or  negative  quantity  accord- 
ing as  M  is  to  the  right  or  left  of  G. 
"M    By  Euclid  II,  12,  13,  we  have  OA2 
=  OG2  +  Ga2  +  2OG.*;  therefore, 

m  .  OA2  =  m  .  OG2  +  m  .  GA2  +  2OG  .  m  .  x. 

A  similar  result  holds  good  with  respect  to  every  particle  of 
the  body.  Hence  we  see  that  the  moment  of  inertia  with 
respect  to  the  assigned  axis  is  composed  of  three  parts, 
namely,  first  the  sum  of  such  terms  as  w.OG2,  and  this  will 
be  equal  to  the  product  of  the  mass  of  the  body  into  OG2; 
secondly,  the  sum  of  such  terms  as  m.Ga2,  and  this  will  be 
the  moment  of  inertia  of  the  body  about  the  axis  through 
G;  and  thirdly  the  sum  of  such  terms  as  2OG .m.xy  which  is 
zero.  Hence  the  moment  of  inertia  about  the  assigned  axis 
has  the  value  stated  in  the  proposition. " 

Be  it  remembered  that  this  book  is  a  book  for  "beginners". 
I  remember  a  Fourth  Form  once  being  given  ten  minutes 
to  read  up  the  subject-matter  just  quoted.  Then  came 
questions.  Said  one  boy,  "  I  thought  inertia  meant  lazi- 
ness. " — "  So  it  does,  a  sort  of  laziness."—"  Then  does 
'  moment  of  inertia  '  mean  a  moment  of  laziness?"  Said 
another  boy,  "  How  are  we  to  find  the  mass  of  one  particle? 
Do  we  crush  the  thing  up  in  a  mortar,  and  weigh  one  of  the 
particles?  or  do  we  weigh  the  thing  first,  then  crush  it  up, 
count  up  the  particles,  and  divide  the  weight  by  the  number?" 
The  teacher  replied,  "  Don't  be  silly;  moment  of  inertia  is 
not  real;  it  is  only  theory  "! 

Who  could  blame  the  boys  for  asking  such  questions? 
How  could  they  have  obtained  the  faintest  insight  into  the 
nature  of  the  subject  under  discussion? 

Forty  or  fifty  years  ago,  Todhunter's  Analytical  Statics 
was  a  standard  work,  used  by  mathematical  students  at  the 


MECHANICS  493 

University.  There  are  cases  on  record  of  men  who  obtained 
Firsts  in  mathematics  but  who  in  the  subject  mentioned  had 
read  no  other  book  at  all,  had  never  handled  a  piece  of  appa- 
ratus in  their  lives.  Fortunately  that  age  has  passed  away. 

Mr.  Fletcher's  trolley,  which  is  now  in  general  use  for 
teaching  dynamics,  is  not  always  made  so  serviceable  as  it 
might  be.  (Readers  should  refer  again  to  Mr.  Fletcher's 
own  comprehensive  article  in  the  School  World  for  1904.) 
In  Perry's  Teaching  of  Elementary  Mechanics,  already  referred 
to,  Mr.  Ashford,  formerly  Head  of  the  Royal  Naval  College, 
Dartmouth,  gives  some  exceedingly  useful  hints  on  the 
further  use  of  the  trolley. 

The  early  teaching  of  mechanics  must  be  given  an  ex- 
perimental basis.  Mathematicians  unacquainted  with  the 
mechanical  laboratory  should  let  the  subject  alone.  It  is 
better  not  taught  at  all  than  to  be  taught  as  mere  algebra  and 
geometry.  Only  if  basic  principles  are  established  experi- 
mentally can  the  subsequent  mathematical  work  be  given 
a  reality  and  a  rigour  that  command  respect. 

44  The  Teaching  of  Mechanics  in  Schools  " 

A  report  on  "  The  Teaching  of  Mechanics  in  Schools  ", 
specially  prepared  for  the  Mathematical  Association,  was 
issued  in  1930.  The  responsible  sub-committee  was  appointed 
in  1927  by  the  General  Teaching  Committee  of  the  Association. 
The  sub-committee  included  such  well-known  teachers  as 
Mr.  C.  O.  Tuckey,  Mr.  W.  C.  Fletcher,  Mr.  W.  J.  Dobbs, 
Mr.  C.  J.  A.  Trimble,  and  Mr.  A.  Robson,  and  the  Report 
will  therefore  carry  great  weight  amongst  all  teachers  of 
mathematics.  Every  page  reveals  the  hand  of  the  practical 
teacher.  No  teacher  of  mathematics  should  fail  to  give  it 
his  serious  attention.  We  quote  a  few  short  paragraphs  in 
order  that  the  reader  may  gather  some  notion  of  the  general 
tenor  of  the  Report. 

"  There  is  perhaps  no  branch  of  mathematical  instruction 
for  which  a  pupil  comes  prepared  with  a  larger  body  of 


494  CRAFTSMANSHIP  IN  MATHEMATICS 

intuitional  knowledge  than  he  does  for  mechanics.  The 
suggestions  made  in  this  report  are  based  on  the  view  that 
this  body  of  knowledge  should  form  the  foundation  of  the 
teaching,  and  that  the  aim  of  the  teaching  should  be  largely 
concerned  with  a  development  of  a  taste  for  such  accurate 
thought  and  consideration  of  mechanical  facts  as  will  make 
them  more  intelligible,  increasing  the  interest  which  attaches  to 
the  mechanical  behaviour  of  things,  and  leading  to  that  insight 
which  brings  this  behaviour  more  completely  under  control/' 

"  Just  as  geometry  has  its  roots  in  familiar  phenomena  of 
daily  life,  so  has  mechanics.  The  basic  principles  of  both 
sciences  can  be  gathered,  at  least  crudely,  from  ordinary 
observation — this  is  the  process  we  knew  as  abstraction. " 

"  When  we  have  carried  the  process  some  little  way  it 
becomes  necessary,  or  at  least  economical,  to  arrange  things 
so  as  to  provide  a  more  exact  answer  to  a  definite  question 
than  can  be  obtained  from  observation  of  unarranged  or 
uncontrolled  phenomena.  So  we  get  two  processes,  fading 
into  one  another  no  doubt  in  marginal  cases,  but  in 
general  easily  distinguishable,  viz.  reflection  on  ordinary 
experience,  and  deliberately  arranged  experiment.  In  the 
former  it  may  be  noted — and  it  is  perhaps  an  essential  part 
of  the  distinction — experience  comes  before  thought;  we 
may  or  may  not  observe  and  reflect  upon  it  and  we  may  or 
may  not  make  scientific  use  of  it.  In  the  latter,  viz.  experi- 
ment, as  it  has  to  be  deliberately  arranged,  thought  comes 
first — we  must  frame  a  question  before  we  can  arrange  the 
experiment  which  is  to  give  the  answer. " 

"  While  there  is  room  for  difference  of  opinion  and 
practice  as  to  the  place  of  experiment  in  the  school  treatment 
of  mechanics,  there  is  no  question  that  observation  and 
reflection  on  ordinary  experience  are  essential  for  any  proper 
grasp  of  the  subject.  The  widespread  neglect  of  this  obvious 
truth  is  responsible  for  much  lack  of  success  in  the  teaching 
of  the  subject." 

"  In  mechanics  the  crude  facts  lie  open  to  direct  observa- 
tion, and  the  role  of  experiment  is  limited  to  rendering  more 


MECHANICS  495 

precise  an  answer  which,  in  the  rough,  can  be  given  without 
experiment." 

"  The  function  of  experience  is  to  provide  a  basis  of  reality 
for  the  abstract  science  of  the  textbook  and  the  schoolmaster, 
and  the  paramount  duty  of  the  latter  is  to  make  his  pupils 
conscious  of  their  own  experience,  to  get  them  to  reflect 
upon  it,  to  co-ordinate  their  existing  store  and  to  open  their 
eyes  to  observe  more  closely  and  to  see  the  significance  and 
interest  of  much  that  the  unobservant  mind  ignores.  Training 
of  this  sort  is  essential  if  the  subject  is  to  have  its  real 
value.  ...  In  each  fresh  section  of  the  work,  the  first  thing 
to  do  is  to  collect  and  clear  up  existing  experience  bearing 
on  the  matter  in  hand." 

The  "  Contents  "  of  the  Report  are  as  follows: 

1.  Position  in  the  Curriculum. 

2.  General  Aims. 

3.  Experience  and  Experiment. 

4.  Order  of  Treatment. 

5.  The  beginning  of  Statics. 

6.  The  beginning  of  Dynamics. 
7    Miscellaneous  Topics: 

(i)  Earlier  teaching  of  Mechanics;  (ii)  Experiments;  (iii) 
Initial  difficulty  of  Statics;  (iv)  Kinematics;  (v)  Units  and 
Dimensions;  (vt)  Horse  Power;  (vii)  Formation  of  the 
Equations  of  Motion;  (viii)  Jointed  Frames;  (ix)  Friction; 
(x)  Torque,  Couples;  (xi)  Geometrical  and  Algebraic 
Methods;  (xii)  Impact  and  the  Lew  of  Momentum 
and  Energy;  (xiii)  Rotatory  Motion;  (xiv)  Limitations 
of  School  Dynamics;  (xv)  Miscellaneous. 

8.  To  examiners. 

9.  Appendices: 

(i)  Wheeled  Vehicles;    (ii)  Momentum  Diagram. 

Newtonian  Mechanics  superseded 

It  is  commonly  said  that  Einstein  has  dethroned  Newton, 
and  this  in  a  sense  is  true,  inasmuch  as  Newton's  laws 
have  been  superseded;  but  Einstein  has  always  regarded 
Newton  as  his  master.  Improved  instruments  have  led  to 


496  CRAFTSMANSHIP  IN  MATHEMATICS 

the  discovery  of  facts  unknown  to  Newton,  and  Newton's 
laws  have  had  to  be  amended  in  order  that  the  new  facts 
may  be  included,  and  this  has  been  really  Einstein's  work. 

At  the  end  of  last  century,  physical  science  recognized 
three  indisputable  universal  laws:  (1)  conservation  of  matter; 
(2)  conservation  of  mass;  (3)  conservation  of  energy;  and 
on  the  strength  of  these  laws  physical  science  became  almost 
aggressively  dogmatic.  They  should,  of  course,  have  been 
regarded  merely  as  working  hypotheses.  Since  1905,  it  has 
been  recognized  that  energy  of  every  conceivable  kind  has 
mass  of  its  own.  Mass  is  the  aggregate  of  rest-mass  and 
energy-mass.  Mass  is  seen  to  be  conserved  only  because 
matter  and  energy  are  conserved  separately. 

Then,  again,  as  to  the  question  of  fixed  axes.  The  trouble 
that  some  of  us  had  when  learning  mechanics  in  the  days 
of  our  youth  arose  (as  we  now  see)  from  the  assumption  that 
axes  were  fixed  in  space.  It  is  impossible  not  to  feel  that 
such  able  men  as  Kelvin,  Tait,  and  Routh  were  not  suspicious 
that  the  theory  was  in  some  way  incomplete,  but  they  seem 
to  have  acquiesced  in  giving  to  Newton's  laws  of  motion 
a  universality  and  finality  which  we  now  know  the  laws 
did  not  really  possess. 

Listen  to  Clerk  Maxwell  (as  a  mathematician  probably 
second  only  to  Newton),  in  his  lighter  moments: 

"RIGID   BODY   (Sings) 

"  Gin  a  body  meet  a  body 

Flyin'  through  the  air, 
Gin  a  body  hit  a  body 

Will  it  fly?  and  where? 
Ilka  impact  has  its  measure, 

Ne'er  a  ane  hae  I, 
Yet  a*  the  lads  they  measure  me 

Or,  at  least,  they  try. 

"  Gin  a  body  meet  a  body 

Altogether  free, 
How  they  travel  afterwards 
We  do  not  always  see. 


MECHANICS  497 

Ilka  problem  has  its  method 

By  analytics  high; 
For  me,  I  ken  na  ane  o*  them, 

But  what  the  waur  am  I?" 

How  are  the  tremendously  far-reaching  twentieth  century 
changes  to  affect  our  teaching?  Probably  not  at  all  except 
in  the  Sixth  Form,  for  another  twenty  years  to  come.  Of 
course  the  changes  are  very  slight,  too  slight  to  affect  appreci- 
ably the  actual  practice  of  mechanics.  But  the  theory  of 
mechanics  is  another  story  altogether. 

Books  to  consult: 

1.  Mechanics,  J.  Cox. 

2.  Introduction  to  the  Principles  of  Mechanics,  J.  F.  S.  Ross. 

3.  Theoretical  Mechanics,  J.  H.  Jeans. 

4.  Mechanics  of  Fluids,  E.  H.  Barton. 

5.  Treatise  on  Hydrostatics,  G.  W.  Minchin. 

Routh,  and  Lamb,  should  still  be  on  every  teacher's  shelf.  Elemen- 
tary books  like  Ashford,  Eggar,  and  Fawdry,  are  full  of  useful  teaching 
hints.  The  book  for  every  teacher  to  master  is  Science  of  Mechanics 
(Mach). 


CHAPTER  XXXIII 

Astronomy* 

Mathematics  or  Physics? 

If  astronomy  is  included  in  the  school  physics  course, 
the  necessary  mathematical  work  will  be  mainly  supplementary. 
If  the  subject  has  to  be  included  wholly  in  the  mathematical 
course,  it  is  not  likely  to  have  any  great  value.  Mathematical 
astronomy  which  is  not  based  upon  personal  observations 

*This  chapter  should  be  read  in  conjunction  with  Chapter  XXVI  of  Science 
Teaching. 

(E291)  33 


498  CRAFTSMANSHIP  IN  MATHEMATICS 

of  any  kind,  with  the  telescope  at  least,  if  not  with  the 
spectroscope,  is  not  likely  to  have  much  reality. 

Elementary  Work 

A  certain  amount  of  introductory  astronomy  will  neces- 
sarily be  included  in  a  school  geography  course.  For 
instance: 

1.  The  earth  as  a  globe  travelling  round  the  sun  and 
spinning  all  the  time  on  its  own  axis  inclined  661°  to  the 
plane  of  the  ecliptic,  i.e.  the  plane  of  its  path  round  the  sun. 

2.  The  consequences  of  these  movements:   day  and  night, 
the  seasons. 

3.  The  moon  as  a  globe  spinning  on  its  own  axis  once  a 
month,  and  travelling  round  the  earth  once  a  month,  in  a  plane 
slightly  inclined  to  the  plane  of  the  ecliptic.     Phases  of  the 
moon. 

4.  Eclipses:    comparative  rarity  of  the  phenomenon  the 
result  of  the  inclination  of  the  orbits  of  the  earth  and  moon. 

5.  Fixing  positions  on  the  earth's  surface.   Latitude  and 
longitude.    Elementary  notions  of  map  projection. 

Older  pupils  who  have  done  a  fair  amount  of  geometry, 
especially  geometry  of  the  sphere,  have  no  difficulty  in  under- 
standing these  things  from  descriptions  and  diagrams.  But 
younger  pupils  require  more  help,  otherwise  they  cannot  visu- 
alize the  phenomena,  they  remain  puzzled,  and  their  written 
answers  to  questions  are  seldom  satisfactory. 

If  an  orrery  is  available,  there  is  little  difficulty,  but  more 
often  than  not  the  teacher  has  to  manage  with  improvised 
models,  perhaps  a  mounted  globe  to  represent  the  earth,  and 
painted  wooden  balls  to  represent  the  sun  and  moon.  Per- 
sonally I  prefer  to  use  a  large  porcelain  globe  (the  kind  used 
with  the  old-fashioned  paraffin  lamps)  to  represent  the  sun, 
the  globe  being  fixed  in  position  a  foot  or  so  above  the  centre 
of  the  table,  and  illuminated  from  the  inside  by  the  most 
powerful  electric  light  available,  the  room  being  otherwise  in 


ASTRONOMY  499 

darkness.  This  makes  an  admirable  sun,  and  gives  a  sharply 
defined  shadow.  The  earth  may  be  represented  by  a  small 
wooden  ball  painted  white,  with  a  knitting-needle  thrust 
through  its  centre  to  represent  the  axis,  and  with  black  circles 
to  represent  the  equator  and  the  23^°  and  66^°  parallels,  the 
ball  being  mounted  so  that  its  centre  is  the  same  height 
above  the  table  as  is  the  centre  of  the  sun,  and  the  axis  being 
inclined  at  66£-°.  About  one-half  the  "  earth  "  is  now  brilliantly 
illuminated,  and  the  other  half  is  in  shade.  If  the  earth  is 
moved  round  in  its  orbit,  the  successive  positions  of  its  axis 
maintaining  a  constant  parallelism,  the  meaning  of  (i)  day 
and  night  and  their  varying  length  in  different  parts  of  the 
world,  and  (ii)  the  seasons,  may  be  made  clear  in  a  few 
sentences.  If  more  serious  work  is  to  be  done  later,  it  is 
particularly  necessary  that  the  plane  of  the  ecliptic  should 
be  clearly  visualized,  and  this  is  easily  done  if  the  sun  and  the 
earth  are  supposed  to  be  half  immersed  in  water,  the  surface 
of  the  water  representing  the  plane  of  the  ecliptic.  Make 
the  pupils  see  clearly  that  half  the  earth's  equator  is  always 
above,  and  the  other  half  always  below,  this  plane. 

The  phases  of  the  moon  are  best  taught  by  ignoring  the 
model  of  the  earth  for  the  time  being  and  considering  models 
of  the  sun  and  moon  alone.  Let  the  laboratory  sun  illuminate 
a  painted  ball,  to  represent  the  moon;  let  the  pupils  move 
round  this  ball,  from  a  position  where  they  see  the  non- 
illuminated  half  to  the  position  where  they  see  the  fully- 
illuminated  half.  One  "  phase  "  after  another  comes  into 
view,  and  further  teaching  is  unnecessary.  Now  put  the 
"earth"  in  position,  and  show  how  the  earth  may  get  between 
the  sun  and  the  moon,  and  prevent  the  sun  from  shining 
on  the  moon;  and  how  the  moon  may  get  between  the  earth 
and  the  sun,  and  prevent  our  seeing  the  sun.  And  thus  we 
come  to  eclipses. 

The  first  essential  in  teaching  eclipses  is  to  make  pupils 
realize  that  a  cone  of  shadow  is  a  thing  of  three  dimensions. 
Let  the  school  sun  cast  the  shadow  of  the  much  smaller 
school  earth.  The  whole  classroom  remains  brilliantly  lighted 


500  CRAFTSMANSHIP  IN  MATHEMATICS 

save  for  a  cone  of  darkness  on  the  far  side  of  the  earth  (we 
ignore  all  other  objects  in  the  room),  and  the  shape  and 
size  of  this  cone  is  easily  demonstrated  by  holding  a  screen 
at  varying  distances  behind  the  earth.  With  a  second  ball 
to  represent  the  moon,  correct  notions  of  total,  annular,  and 
partial  eclipses  may  be  readily  given.  It  is  quite  easy  to 
show  why  eclipses  are  comparatively  rare  phenomena  by 
making  the  moon  move  round  in  an  orbit  inclined  to  the 
earth 's  orbit. 

More  Advanced  Work 

A  Sixth  Form  ought  to  carry  the  subject  very  much 
farther  than  the  elementary  aspects  of  it  commonly  included 
in  a  geography  course,  but  the  business  of  the  mathematical 
teacher  is  not  to  give  astronomy  lectures  in  the  wider  sense 
but  to  teach  boys  to  solve  those  problems  which  are  suggested 
by  the  results  of  actual  observation;  for  instance,  the  problem 
of  fixing  the  positions  of  the  stars  by  means  of  their  co- 
ordinates, the  related  question  of  the  diurnal  revolution  of 
the  heavens,  the  daily  movements  of  the  sun  and  moon, 
the  calculation  of  times  of  rising  and  setting,  nautical  problems 
of  determining  latitude  and  longitude,  dialling  problems. 

Facts  must  not  be  confused  with  hypotheses.  Thus  the 
earth's  daily  rotation  on  its  axis  and  its  annual  revolution 
round  the  sun  are  mere  hypotheses,  invented  to  account  for 
facts  of  observation.  The  mathematical  teacher  is  concerned 
with  the  face  value  of  the  facts  observed.  According  to  that 
face  value,  the  stars  move  round  the  sky  daily,  and  the  sun 
and  moon  move  amongst  them.  Any  attempt  to  provide 
a  theory  of  stellar  movements  must  be  preceded  by  an  exact 
determination  of  the  facts  as  they  appear. 

Quite  low  down  the  school  the  boys  ought  to  have  been 
made  familiar  with  the  globe  (a  blackboard  surface  is  very 
useful)  and  a  cardboard  horizon  fitting  over  it.  And  in  the 
very  early  stages  of  geometry  they  will  have  been  introduced 
to  the  theodolite,  and  will  have  been  taught  to  measure 
altitudes  and  azimuths  (though  perhaps  the  term  azimuth 


ASTRONOMY  501 

has  not  been  used).  The  theodolite  may  have  been  made  in 
the  school  workshop,  and  a  mere  cardboard  tube  used  instead 
of  a  telescope.  But  higher  up  the  school  an  instrument 
designed  for  fairly  accurate  measurements  should  be  used, 
and  nowadays  a  good  one  may  be  purchased  for  a  few  pounds. 
Even  Fourth  Form  boys  can  be  taught  to  measure  the  azimuth 
and  altitude  of  a  given  star  as  it  appears  to  an  observer  at 
a  given  moment.  It  is  easy  and  interesting  work  and  they 
like  it,  though  some  of  them  seem  to  need  repeated  help 
with  the  setting  up  and  initial  adjustment  of  the  instrument. 
I  have  known  boys  of  9  or  10  readily  pick  out  the  better 
known  constellations,  and  such  stars  as  the  Pole  Star,  Vega, 
Capella,  Sirius,  and  the  Pleiades.  This  kind  of  observation  work 
ought  to  be  included  in  every  Nature  Study  course.  It  creates 
an  early  interest  that  becomes  permanent,  and  such  basic 
facts  are  very  useful  for  future  mathematical  work. 

A  school  lucky  enough  to  have  a  small  observatory  of 
its  own  will  have  an  altazimuth  (a  theodolite  is  virtually 
a  portable  altazimuth),  so  that  azimuths  and  altitudes  (or 
zenith  distances)  may  readily  be  found.  An  equatorial  may 
also  be  available.  If  not,  the  altazimuth  should  be  of  such 
a  kind  that  its  telescope  can  be  mounted  equatorially  when 
required.  Then  the  boys  can  take  Declinations  and  Right 
Ascensions,  and  become  familiar  with  the  celestial  equator 
as  well  as  with  the  celestial  pole,  and  they  will  then  soon 
look  upon  the  rotating  northern  celestial  hemisphere  as  an 
old  familiar  friend.  Once  they  feel  this  familiarity,  the 
making  of  reasonably  accurate  observations  is  child's  play 
and  the  mathematics  involved  is  not  difficult.  The  sidereal 
clock  and  sidereal  time  are  also  easily  mastered. 

The  solution  of  the  common  problem  of  determining  the 
altitude  and  azimuth  of  a  star  when  the  hour- angle  and 
declination  are  given  (or  vice  versa)  is  an  easy  case  of  the 
solution  of  a  spherical  triangle,  and  should  be  familiar. 

The  sun-dial  cannot  profitably  be  taken  up  until  the 
Sixth,  and  not  even  then  unless  the  boys  have  been  well 
grounded  in  the  geometry  of  the  sphere  and  its  circles.  The 


502  CRAFTSMANSHIP  IN  MATHEMATICS 

geometrical  method  of  graduating  the  dial  (to  be  fixed  either 
horizontally  or  on  a  south  wall)  is  simple  enough  if  the 
elementary  geometry  of  the  sphere  has  been  mastered.  The 
boys  must  be  able  to  see  that  the  key  to  the  whole  thing  lies  in 
the  fact  that  the  edge  of  the  gnomon  is  parallel  to  the  earth's 
axis  and  is  therefore  pointing  in  the  direction  of  the  celestial 
pole.  If  about  this  they  are  vague,  the  whole  thing  is  vague. 

There  is  no  better  way  of  introducing  the  young  observer 
to  the  knowledge  of  the  law  of  the  sun's  rotation  than  by 
leading  him  to  see  that,  if  a  dial  be  so  placed  that  the  style 
(the  edge  of  the  gnomon)  is  parallel  to  the  axis  of  the  rotating 
celestial  hemisphere,  the  shadow  of  the  style  will  at  all  seasons 
of  the  year  move  uniformly  over  the  receiving  surface  at  the 
rate  of  15°  an  hour. 

The  graduation  of  a  sun-dial  to  be  placed  on  a  vertical 
wall  is  not  difficult,  but  it  is  a  good  little  puzzle  for  testing  a 
boy's  knowledge  of  the  sphere  and  his  powers  of  visualizing 
the  true  geometrical  relations  of  the  parts  of  a  rather  com- 
plicated figure. 

Mathematical  problems  in  astronomy  are,  of  course,  un- 
limited, but  in  school  there  is  no  time  to  touch  upon  more 
than  the  bare  fundamentals. 

Whitaker's  Almanack  is  a  mine  of  useful  data  for  problem 
purposes. 

Stellar  Astronomy 

The  main  interest  of  astronomers,  and  indeed  that  of  the 
general  public,  is  now  concerned  with  the  stars  and  nebulae 
rather  than  with  the  solar  system."  With  the  main  facts  of 
the  solar  system  every  boy  should  be  made  familiar;  but 
stellar  astronomy  is  more  difficult,  the  greater  part  of  the 
available  evidence  being  merely  of  an  inferential  character. 
In  a  very  large  measure  we  have  to  deal  with  probabilities, 
not  certainties. 

The  astronomer's  principal  instruments  are  the  telescope 
(mounted  in  different  ways  according  to  the  work  to  be  done), 
the  spectroscope,  the  camera,  and  the  interferometer.  The 


ASTRONOMY  503 

last-named  is  outside  possible  school  practice,  so  is  the 
camera.  But  the  spectroscope  is  now  in  common  use  in 
schools,  and  as  it  ranks  next  to  the  telescope  in  the  work 
of  an  observatory,  its  uses  should  be  taught  thoroughly. 

A  course  of  instruction  may  be  expected  to  include  the 
following: 

1.  Spectrum  analysis.    Displacement  of  lines:  the  causes; 
difficulty  of  interpretation;    distance  and  speed  effects  con- 
sidered separately. 

2.  The  galactic  system  of  stars. 

3.  The  extra-galactic  system:   stars  and  nebulae. 

4.  Stellar  spectra.     Interpretation  of  photographs. 

5.  Stellar   magnitudes,   movements,  velocities,  distances, 
temperatures;    how  determined. 

6.  Theories  of  stellar  structure:   for  instance,  (i)  Edding- 
ton's,  (ii)  Jeans'. 

7.  Solar    radiation.       Energy    and    temperature    of   sun. 
Poincare's  theorem. 

8.  Stellar  radiation  and  cosmic  radiation  generally.    Hoff- 
mann's determination  of  the  sun's  contribution  to  the  total 
cosmic  ultra-radiation;  inferences  therefrom.     Hess's  views. 

9.  Relativity.    General  outline.    Einstein's  proposed  tests. 
Confirmation  of  the  tests  and  final  acceptance  of  the  theory. 

10.  Modern   cosmologies:    (i)  Einstein's,  (ii)  De   Sitter's. 
Do  they  clash?    Lemaitre's  views — how  an  Einstein  universe 
may  expand  to  a  De  Sitter  universe. 

11.  Rival  theories  as  to  the  future  of  the  universe.    British 
physicists'  views  of  a  universe  slowly  running  down  to  a  state 
of  thermodynamic  equilibrium.   Millikan's  views  of  a  universe 
being  continually  rebuilt.    Evidence  pro  and  con. 

How  much  of  this  work  will  be  done  by  the  mathe- 
matical teacher?  His  task  will  probably  be  concerned  mainly 
with  two  things:  (i)  some  easy  but  extremely  interesting 
arithmetic;  (ii)  the  very  difficult  subject  of  Relativity. 

Mathematical  teachers  differ  in  opinion  as  to  the  wisdom 
(or  folly)  of  introducing  relativity  in  a  Sixth  Form  course. 


504  CRAFTSMANSHIP  IN  MATHEMATICS 

But  in  view  of  the  far-reaching,  indeed  fundamental,  changes 
that  the  subject  is  bringing  about  in  the  whole  domain  of 
physics,  it  seems  desirable  that  an  attempt  should  be  made 
to  give  Sixth  Form  specialists  at  least  an  outline  of  the  subject. 
After  all,  the  "  special  "  theory  of  relativity  is  easily  taught, 
and,  this  done,  the  much  more  difficult  "  general  "  theory 
may  be  so  far  touched  upon  that  the  final  results  of  the  theory 
may  be  fairly  well  understood  by  the  abler  boys.  Professor 
Rice's  and  Mr.  DurelPs  little  books  may  be  followed  up 
by  Einstein's  own  elementary  book,  and  his  by  Nunn's 
Relativity  and  Gravitation,  which  is  by  far  the  best  book 
on  the  subject  from  the  teacher's  point  of  view.* 

The  arithmetic  of  stellar  astronomy  deals  with  numbers 
so  vast  that  it  is  likely  to  deceive  all  but  the  trained  mathe- 
matician. How,  for  instance,  may  we  bring  home  to  a  boy 
the  real  significance  of  the  following: 

1.  The  sun  is  losing  weight  by  radiation  at  the  rate  of 
1-31  . 1014  tons   a   year,  yet    2 .  10°   years   ago   it   was   only 
1-00013  times  its  present  weight. 

2.  Weight  of  sun  —  2  . 1033  grammes. 

3.  Temperature  of  interior  of  sun  =  4  . 108  degrees. 

4.  Number  of  stars  in  galactic  system  =  4 .  1011. 

5.  The    2,000,000    extra -galactic    nebulae    each    contain 
enough  matter  to  make  2 . 109  stars,  that  is  4 . 1015  stars  in  all. 

6.  The  extra-galactic  nebulae  are  at  an  average  distance 
away  of  140  million  light-years  (1  light-year  =  6  .  1012  miles) 
and  their  average  distance  apart  from  each  other  is  of  the 
order  of  2  million  light-years. 

7.  Radius  of  universe  is  perhaps  2000  million  light-years 

=  2  . 109  X  6  . 1012  miles 
=  1-2  .  1022  miles. 

We  shall  refer  to  this  subject  again  in  a  later  Chapter. 

Be  consistent  when  using  the  terms  "  world  ",  "  uni- 
verse ",  "  cosmos  ",  "  space  ",  "  ether  ",  "  space-time  ".  It 

*  For  detailed  suggestions  see  Chapter  XXXII  of  Science  Teaching. 


ASTRONOMY  505 

is  probably  sufficient  to  tell  a  boy  that  the  matter- containing 
universe,  no  matter  how  large,  is  itself  within  a  limitless  void. 
Do  not  let  him  think  that  the  mathematician's  convenient 
and  necessary  fiction  "  space-time  "  is  any  sort  of  glorified 
Christmas  pudding  mixture.  The  mathematical  partnership 
is  purely  formal.  Distinguish  between  an  infinite  void  and 
a  limited  wave-carrying  matter-containing  universe. 

In  his  address  to  the  Mathematical  Association,  January, 
1931,  Sir  Arthur  Eddington  said:  "About  every  1,500,000,000 
years  the  universe  will  double  its  radius  and  its  size  will 
go  on  expanding  in  this  way  in  Geometrical  Progression  for 
ever." — A  rude  boy  might  ask  some  very  awkward  questions 
on  this  point,  and  carry  his  teacher  backwards  as  well  as 
forwards  in  limitless  time.  It  is  of  no  use  merely  to  go  back 
to  an  assumed  initial  state  of  equilibrium.  The  boy  is  certain 
to  say,  and  before  that? 

Books  to  consult: 

In  selecting  books  on  Astronomy,  don't  forget  some  of  the  older 
writers,  e.g.  Herschel,  Proctor,  Lockyer,  Ball.  Eddington 's,  Jeans', 
and  Turner's  books  should  be  known  to  all  teachers  of  mathematics. 
Barlow  and  Bryan's  Elementary  Mathematical  Astronomy  is  very 
useful.  From  the  teacher's  point  of  view,  Sir  Richard  Gregory's 
books  take  quite  the  first  place.  Consult  also  Dingle's  Astrophysics. 

Readers  who  are  specially  interested  in  Relativity  should  read 
Dr.  John  Dougall's  searchingly  critical  article  in  Vol.  X  of  the 
Philosophical  Magazine,  pp.  81-100. 


503  CRAFTSMANSHIP  IN  MATHEMATICS 


CHAPTER  XXXIV 

Geometrical  Optics 

Present  Methods  of  Teaching  often  Criticized 

We  include  this  subject  because  it  quite  properly  belongs 
to  mathematics  as  well  as  to  physics. 

Probably  no  part  of  the  teaching  of  mathematics  or  of 
physics  is  so  severely  criticized  as  the  teaching  of  optics,  no 
matter  whether  the  subject  is  taught  by  the  mathematics  teacher 
or  by  the  physics  teacher.  That  there  is  an  urgent  need  for 
some  reform  will  be  readily  admitted  from  the  discussion 
on  "  The  Teaching  of  Geometrical  Optics  "  that  took  place 
on  April  26,  1929,  reported  fully  on  pp.  258-340  in  No.  229 
of  the  Proceedings  of  the  Physical  Society.  Papers  were  read 
by  a  number  of  persons  interested  in  optics,  including  several 
Public  School  and  University  teachers  and  representatives 
of  the  optical  industry.  A  few  of  the  teachers  tried  to  defend 
the  present  system,  though  not  very  successfully.  The  conflict 
of  opinion  centred  largely  (1),  round  the  place  to  be  given  and 
the  purpose  to  be  assigned,  in  a  teaching  course  of  optics,  to 
the  reciprocal  equation  (1/u  -\~  I/v  —  I/f),  and  (2),  round  the 
question  of  "  rays  or  waves  ".  My  own  quite  definite  con- 
clusion from  the  discussion  was  that  the  best  way  of  teaching 
the  subject  is  to  begin  with  elementary  physical  optics  in 
Forms  IV  and  V,  and  to  defer  geometrical  optics  until  Form  VI. 

Several  of  the  critics  found  fault  with  the  present  system 
because  it  fails  to  supply  a  sufficient  practical  knowledge 
of  optical  instruments  and  their  performance;  because  pupils 
by  the  end  of  their  course  in  optics  have  done  little  more  than 
devote  their  time  to  elementary  algebra  and  geometrical 
diagrams  which  have  but  a  very  slender  relation  to  the  subject 
under  consideration;  because,  in  short,  the  utility  of  the  subject 
is  extremely  meagre. 


GEOMETRICAL  OPTICS  507 

My  own  main  criticism  takes  another  direction — that 
the  mathematics  and  the  theory  of  the  subject  at  present 
tend  to  take  too  early  a  place  in  the  teaching  course,  inasmuch 
as  the  physical  phenomena  underlying  the  mathematics  and 
the  theoretical  arguments  have  not  been  studied,  the  arguments, 
therefore,  having  no  real  significance. 

Rays  or  Waves? 

Hitherto  the  "  ray  "  method  of  teaching  has  been  almost 
universal  in  our  schools,  but  the  mathematics  has  been 
too  much  divorced  from  experiment  and  its  real  significance 
has  been  ill  understood.  In  the  discussion  already  referred 
to,  the  method  was  defended  mainly  because  of  its  simplicity, 
not  because  of  its  practical  utility.  The  protagonist  of  the 
wave  or  curvature  method  was  Dr.  Drysdale,  for  many 
years  head  of  the  optical  department  at  the  Northampton 
Institute,  London.  He  advocated  the  method  on  the  grounds 
(amongst  others)  that  (1)  it  simplifies  the  teaching;  (2)  it 
harmonizes  the  teaching  of  science  with  optical  practice; 
and  (3)  it  leads  naturally  to  higher  physical  optics.  The  real 
advantage  of  the  method  seems  to  be  that  it  places  the  whole 
of  optical  teaching  on  a  physical  basis,  and  leads  naturally 
to  the  study  of  interference,  diffraction,  and  polarization. 
Two  well-known  elementary  books  developing  the  subject 
on  a  wave  basis  are  those  of  Mr.  W.  E.  Cross  and  Mr.  C.  G. 
Vernon. 

Whichever  method  is  used,  the  teacher  should  be  quite 
frank  in  stating  that  energy  can  be  radiated  in  two  forms, 
corpuscles  and  waves.  Both  forms  are  easily  illustrated  experi- 
mentally. For  example,  replicas  of  diffraction  gratings  (if 
gratings  themselves  are  too  expensive  to  buy)  are  suitable 
for  illustrating  the  periodic  character  of  light.  In  fact,  the 
periodic  character  of  light  must  be  experimentally  demon- 
strated in  some  way  before  the  curvature  method  can  logically 
be  introduced,  and  this  means  a  preliminary  study  of  the 
velocity  of  light. 


508  CRAFTSMANSHIP  IN  MATHEMATICS 

It  is  a  good  thing  to  teach  both  methods,  and  to  teach 
them  more  or  less  in  parallel.  A  ray  may,  for  instance,  be 
looked  upon  as  a  line  representing  an  element  of  the  wave- 
front,  or  as  a  normal  to  the  wave-surface;  or  the  wave-front 
may  be  traced  as  a  series  of  arcs  after  the  rays  have  been  drawn 
graphically. 

The  best  defence  of  the  wave  method  is  that  the  whole 
of  physics  is,  fundamentally,  a  study  of  wave  systems,  and 
it  is  therefore  difficult  to  justify  the  picking  out  of  one  branch 
and  treating  it  on  an  entirely  different  basis.  But  the  ob- 
jection to  the  ray  method  largely  disappears  if  the  ray  be 
thought  of  as  an  element  of  a  wave,  and  to  the  lens  designer 
the  ray  is  the  all-important  thing. 

Theories  of  Light 

The  whole  question  turns  largely  on  an  acceptable  theory 
of  light.  But  whose  theory?  Newton's?  Fresnel's?  Young's? 
Maxwell's?  Planck's?  de  Broglie's? 

Newton's  corpuscular  theory  failed  to  account  for  certain 
observed  facts.  The  wave  theory  which  superseded  it  was 
also  found  to  be  defective,  and  to  eliminate  these  defects  the 
"  quantum  "  theory  has  been  devised.  The  new  theory  has 
shown  that  Newton  was  not  wholly  wrong  in  regarding  light 
as  corpuscular,  for  that  theory  is  based  on  the  experimental 
fact  that  a  beam  of  light  may  be  considered  to  be  broken  up 
into  discrete  units  called  "  light-quanta  "  or  "  photons  ", 
"  with  almost  the  definiteness  with  which  a  shower  of  rain 
may  be  broken  up  into  drops  of  water,  or  a  gas  into  separate 
molecules  ".  At  the  same  time,  the  light  preserves  its  undu- 
latory  character.  Each  photon  has  associated  with  it  a  perfectly 
definite  quantity  of  the  nature  of  a  wave-length. 

There  seems  to  be  no  doubt  at  all  that  radiation  of  all 
kinds  can  appear  now  as  waves,  now  as  particles.  But  the 
fundamental  units  of  matter,  electrons  and  protons,  can  also 
appear  now  as  waves,  now  as  particles.  In  many  circumstances 
the  behaviour  of  an  electron  or  proton  is  found  to  be  too 


GEOMETRICAL  OPTICS  509 

complex  to  permit  of  explanation  as  the  motion  of  a  mere 
particle,  and  accordingly  physicists  have  tried  to  interpret  it 
as  the  behaviour  of  a  group  of  waves,  and  in  so  doing  have 
founded  the  branch  of  mathematical  physics  known  as  "  wave- 
mechanics  ". 

In  fact  it  may  be  fairly  said  that  no  single  satisfactory 
theory  of  light  exists  to-day.  The  electromagnetic  theory 
carries  us  a  long  way,  but  in  its  classical  form  it  is  quite 
inadequate  to  carry  us  the  whole  way.  The  powerful  methods 
devised  by  Hamilton  in  geometrical  *  mechanics  and  geome- 
trical optics  are  being  used  to  found  a  wave-mechanics  bearing 
to  geometrical  mechanics  a  relation  similar  to  that  which 
wave-optics  bears  to  geometrical  optics.  The  quasi  light- 
particles  emerge  from  this  mechanics  more  or  less  naturally, 
so  that  we  are  practically  back  to  Newton  and  working  on 
Newton's  lines.  The  two  views  are  blended;  neither  is 
destroyed. 

Geometrical  optics  is  as  worthy  of  serious  study  as  geome- 
trical mechanics.  Each  is  the  limiting  form  when  A  ~>  0, 
and  for  many  purposes  this  limiting  mathematical  form  is 
not  only  entirely  sufficient  but  it  is  vastly  simpler,  mathemati- 
cally, than  the  general  wave-form,  whether  in  optics  or  in 
mechanics.  What  is  not  worthy  of  study  (at  all  events  as 
physics)  is  the  type  of  question  often  set,  the  solution  of  which 
depends  wholly  on  some  mathematical  trick.  Large  numbers 
of  these  are  found  in  such  favourite  old  books  as  Tait  and 
Steele's  Dynamics,  or  Parkinson's  Optics,  or  Heath's  Geometrical 
Optics.  Such  problems  are  possibly  good  as  training  material 
in  mathematics,  but  for  the  display  of  mathematical  talent 
there  is  an  abundance  of  excellent  material  that  is,  in  itself, 
valuable  in  physics  also. 

*  I.e.  Newtonian. 


510  CRAFTSMANSHIP  IN  MATHEMATICS 


The  Teacher  of  Optics 

Should  optics  be  taught  by  the  mathematics  teacher  or 
by  the  physics  teacher?  Admittedly  a  mathematics  teacher 
who  has  had  no  training  in  physics  is  not  likely  to  be  able 
to  appreciate  the  natural  powers  and  limitations  of  optical 
instruments,  or  to  grasp  the  significance  of  certain  matters 
in  optical  theory.  Admittedly,  too,  a  physics  teacher  with 
no  special  knowledge  of  mathematics  will  be  out  of  his  depth 
in  the  Sixth  Form  where,  in  optics,  mathematical  considerations 
count  for  almost  everything,  though  he  will  be  easily  able  to 
cope  with  the  first  considerations  of  the  reciprocal  equation, 
which,  after  all,  is  essentially  a  natural  development  of  fives- 
court  and  billiard-table  geometry.  There  is  thus  very  little 
doubt  about  the  answer  to  our  question.  The  physics  teacher 
should  be  responsible  for  the  physical  optics  in  Forms  IV  and 
V,  and  the  mathematics  teacher  for  the  geometrical  optics 
to  be  done  in  VI.  By  geometrical  optics  is  here  meant  the 
really  serious  mathematical  work  that  should  follow  the  physical 
work,  work  that  is  partly  revisionary  but  mainly  supplementary. 
The  higher  physical  work  in  VI  will,  however,  still  have  to  be 
taken  by  the  physics  teacher. 


Suggested  Elementary  Course:    Mainly  Physics 

This  elementary  course  is  intended  to  be  mainly  experi- 
mental and  to  be  done  in  the  laboratory,  all  consideration 
of  the  theory  of  aberration  being  excluded.  Let  all  serious 
mathematics  and  theoretical  developments  be  postponed  to  VI. 

If  the  wave  method  is  adopted,  wave  motion  and  its  sig- 
nificance will  naturally  be  taught  first.  Of  the  many  wave- 
producing  machines  in  the  market,  select  one,  and  see  that 
the  boys  really  understand  what  it  teaches.  The  propagation 
of  transverse  waves  may  be  shown  by  a  ripple  tank,  illuminated 
stroboscopically,  so  that  the  apparent  rate  of  propagation  may 


GEOMETRICAL  OPTICS  511 

be  slowed  down.  Carry  out  practical  work  with  real  beams  of 
light,  not  by  pin  and  parallax  methods.  The  sunbeam  offers 
a  concrete  starting-point. 

Devote  a  lesson  or  two  to  showing  how  fallible  the  eye 
is  as  a  measuring  instrument,  and  why,  therefore,  instrumental 
aids  are  necessary.  Devise  experiments  to  show  the  limited 
power  of  the  eye  in  unaided  vision,  and  show  the  capacity 
of  the  eye  for  distinguishing  detail  under  different  conditions 
of  illumination  and  size  of  aperture. 

Make  beginners  familiar  with  the  construction  and  use 
of  optical  instruments — the  telescope,  the  microscope,  and 
photographic  lens.  When  a  boy  handles  optical  instruments, 
and  learns  to  adjust,  to  test,  and  to  use  them,  he  acquires 
knowledge  of  their  potentialities  and  limitations;  and  he 
also  becomes  acquainted  with  the  language  of  the  subject. 
Throughout  the  course  keep  in  mind  elementary  notions 
both  of  physiological  optics  and  of  the  psychology  of  vision; 
also  that  the  eye  as  an  optical  instrument  is  very  imperfect, 
deceptive,  and  inconstant.  Teach  beginners  when  using 
optical  instruments  the  importance  of  correct  illumination; 
and  the  uselessness  of  increasing  magnification  beyond  the 
value  suitable  for  the  aperture  actually  effective  in  the  experi- 
ment. Show  that  the  apparent  brightness  of  an  extended  object 
cannot  be  increased  by  optical  means;  the  moon  looks  no 
brighter  through  a  telescope. 

The  key  to  refraction  is,  of  course,  the  mere  retardation 
of  velocity  in  a  denser  medium,  and  the  boys  must  under- 
stand clearly  that  a  refractive  index  is  simply  a  velocity  ratio. 
The  slewing  round  of  the  wave-front  must  be  understood 
to  be  just  a  natural  and  inevitable  consequence  of  any  such 
retardation  and  to  be  applicable  universally  and  not  merely 
in  connexion  with  light.  The  trundling  of  a  garden  roller 
across  a  smooth  lawn  to  a  rough  gravel  drive  affords  a  service- 
able illustration.  If  the  direction  of  motion  across  the  lawn 
is  normal  to  the  line  of  separation  between  grass  and  gravel, 
there  is  merely  retarded  velocity;  if  oblique,  there  is  a  slew- 
ing round  as  well. 


512  CRAFTSMANSHIP  IN  MATHEMATICS 

Suggested  topics: 

1.  Nature  and  propagation  of  light. 

2.  Waves:    motion,    length,    amplitude,    frequency,   velo- 
city. 

3.  Illumination.     Photometry,  especially  the  measurement 
of  illumination  by  daylight  photometer.* 

4.  Experiments  in  brightness,  colour,  persistence  of  vision, 
fatigue,  glare. 

5.  Reflection   and   refraction.      Concept   of  the   ray   as  a 
line  representing  the  direction  of  movement  of  an  element 
of  the   wave-front.      The   use   of  rays   in   optical   diagrams. 
Huygens'  principle. 

6.  Function  of  lenses;   imprinting  of  curvature. 

7.  Interference,  diffraction,  polarization. 

8.  The  spectrum;  the  spectroscope. 

9.  The  spectrometer:  first  considerations. 

10.  The  beginnings  of  mathematics;  the  reciprocal  equation 
as  a  convenient  memorandum  for  elementary  work  at  the 
optical  bench. 

11.  Inverse  square  law;    the  unit  standard  source  of  light, 
the  unit  of  luminous  flux,  the  unit  of  illumination,  and  their 
interrelations. 

Suggested  Advanced  Course:   Largely  Mathematics 

Whatever  books  on  geometrical  optics  teachers  use, 
especially  if  they  are  old  favourites  like  Parkinson  and  Heath, 
it  is  a  good  plan  either  to  compare  these  with  modern  standard 
works  on  the  technical  side  of  the  subject,  or  to  discuss  them 
with  a  friend  acquainted  with  the  optical  industry.  The  im- 
portant thing  is  to  find  out  if  the  principles  laid  down  in  a  book 
will  work. 

•An  examiner  reports  that  at  a  recent  university  examination  he  set  a  simple 
question  on  the  measurement  of  daylight  illumination.  Hardly  any  of  the  240 
candidates  gave  a  complete  answer.  A  common  plan  was  to  balance  sunlight 
against  an  electric  lamp,  using,  say,  a  grease  spot,  assume  the  sun  to  be  93,000,000 
miles  away,  assume  the  inverse  square  law,  and  to  calculate  the  candle-power  of  the 
sun! 


GEOMETRICAL  OPTICS  513 

There  is  no  excuse  whatever  for  teaching  the  subject 
by  methods  that  are  out  of  harmony  with  applied  optics. 
Young  computers  who  are  taken  on  at  optical  works  often 
find  to  their  disgust  that  their  school  and  textbook  knowledge 
is  valueless,  and  they  have  to  be  taught  anew  by  technical 
experts. 

The  commonest  mistake  in  optical  teaching  is  due  to 
the  misuse  or  to  the  misunderstanding  of  the  sign  convention 
and  notation.  This  is  unaccountable,  as  the  convention 
is  the  result  of  international  agreement.  In  the  optical  dis- 
cussion already  referred  to,  an  examiner  said  that  a  year  or 
two  ago  he  marked  250  scripts  in  the  Higher  Certificate 
examination,  the  candidates  having  been  taught  in  schools 
in  different  parts  of  the  country.  In  the  Light  paper  was  a 
simple  question  on  a  lens,  and  247  of  the  candidates  attempted 
it,  but  only  7  of  the  247  obtained  the  correct  result.  Such  a 
record  of  muddleheadedness  is  utterly  inexcusable. 

Remember  that  the  basis  of  all  lens  work  calculation  should 
be  the  deviation  in  a  ray  at  each  surface.  Suppose  that  a  ray 


which  diverges  from  the  point  A  at,  say,  15°  is  to  emerge  from 
the  lens  system,  S,  parallel  to  the  axis.  Since  the  whole 
deviation  is  to  be  15°,  and  if  there  are,  say,  two  surfaces, 
are  the  two  partial  deviations  to  be  7|°  each,  or  in  some 
other  proportion?  What  are  the  criteria  for  what  is  best? 
What  are  the  aberrations?  And  so  on. 

After  a  few  simple  calculations  on  a  simple  lens  for  actual 
wide-angle  cases,  the  boys  will  soon  find  that  the  rule  given 
by  Parkinson  for  the  relative  radii  of  the  surfaces  is  by  no 

(B291)  34 


514  CRAFTSMANSHIP  IN  MATHEMATICS 

means  always  right;  it  is  only  right  when  a  ^  10°  (about), 
while  in  many  lenses  a  is  very  much  greater. 

Wave  optics  must  not,  of  course,  be  forgotten.  For  instance, 
the  wave  equation  and  its  simple  solution  should  be  included. 

Suggested  topics: 

1.  Geometrical  optics:    the  reciprocal  equation  more  fully 
considered.    "  Wave  "  proofs  and  "  ray  "  proofs  compared. 

2.  The  dioptre,  sagitta  (sag),  focal  power.     Show  that  the 
curvature  of  a  wave-front  or  surface  is  measured  by  the  reciprocal 
of  the  radius;    the  surface  with  a  radius  of  1  m.  is  chosen 
as  a  standard.     Rdioptres  =  l/^metres-    Exhibit  a  curve  of  1  m. 
radius  so  that  the  curvature  may  be  visualized.    Point  out  that 
for  a  chord  of  8-95  cm.  the  curvature  in  dioptres  is  represented 
by  the  sag  in  mm.    The  application  of  Euclid,  III,  35,  to  the 
sag.    The  dioptre  spherometer. 

3.  The  ideal  lens  contrasted  with  the  actual  lens.     (The 
solution  of  problems  arising  out  of  actual  lenses  will  in  general 
be  too  difficult.) 

4.  Lenses;    spectacles.      How  the   optician  is   concerned 
with  the  forms  of  lenses  as  well  as  with  their  power. 

5.  Combination  of  lenses  with  prisms  to  correct  defects 
of  convergence  in  the  eyes. 

6.  Thin  lenses  in  contact. 

7.  Lens  combinations. 

8.  Axial  displacement. 

9.  Chromatic  aberration. 

10.  Spherical  aberration:  the  disc  of  confusion. 

11.  Astigmatism,  coma,  distortion. 

12.  Photometry  further  considered.     How  the  distance  of 
star  clusters  and  spiral  nebulae  have  been  determined  by  the 
measurement  of  the  apparent  brightness  of  Cepheid  variables 
contained  therein. 

13.  Modern    instruments;     the    telephoto    lens,    range- 
finders,  prism  binoculars,  kinema  projectors. 

14.  The  more  elementary  considerations  of  such  subjects 
as  defects  of  images,  collineation  between  object  space  and 
image  space,  the  optical  sine  theory,  design  of  instruments. 


GEOMETRICAL  OPTICS  515 

The  correction  of  aberrations  by  calculation  will,  in 
general,  be  too  difficult;  so  will  the  higher  order  of  aberrations 
considered  by  ray-tracing,  though  some  notion  of  ray-tracing 
should  certainly  be  given.  The  general  theory  of  lenses  will 
also  be  too  difficult.  In  short,  a  good  deal  of  this  work  is 
more  suitable  for  the  university  than  for  the  school.  Much 
will  depend  upon  the  close  collaboration  of  the  mathematics 
and  physics  staff.  The  two  aspects  of  the  subjects  must  be 
considered  together. 

The  real  value  of  mathematical  work  in  optics  lies  in  the 
discovery  of  the  general  principles  underlying  the  actual 
behaviour  of  real  optical  systems,  as  contrasted  with  the  imagined 
behaviour  of  ideally  perfect  systems. 

Technical  Optics 

Few  teachers  are  familiar  with  technical  optics.  Very 
few  have  seen  even  the  ordinary  operation  of  grinding  a 
lens.  As  for  the  designing  of  lenses  for  special  purposes, 
or  the  art  of  producing  optical  glass,  it  is  known  to  very 
few  persons  indeed.  Few  teachers  realize  that  for  ordinary 
industrial  purposes  the  index  of  a  glass  is  not  considered 
known  unless  its  value  is  obtained  to  the  fourth  decimal 
place,  and  for  dispersion  to  the  fifth. 

Formerly  when  an  optical  system  had  been  designed,  the 
material  prepared  by  the  designer  was  handed  over  to  a 
number  of  computers  expert  in  the  use  of  logarithmic  tables. 
But  calculating  machines  are  now  used,  to  the  operators  of 
which  the  computation  of  the  elements,  individually  and  in 
combination,  of  the  new  optical  system  is  entrusted.  These 
operators  need  have  no  special  mathematical  equipment, 
other  than  that  of  a  common  knowledge  of  simple  trigo- 
nometrical expressions.  Particular  rays  are  traced  step  by 
step  through  surface  after  surface  for  the  purpose  of  de- 
termining at  various  stages  the  longitudinal  and  transverse 
aberrations.  These  values  are  assessed  by  the  skilled  com- 
puter, who  decides  at  what  particular  part  of  the  system  a 


516  CRAFTSMANSHIP  IN  MATHEMATICS 

modification  can  best  be  effected.  His  special  skill  is  practical, 
the  outcome  of  active  practice  in  the  industry  itself.  It 
involves,  above  all,  good  judgment  in  the  balancing  of 
one  type  of  aberration  against  another,  for  no  optical 
system  can  be  free  from  all  kinds  of  aberration. — Consider 
the  amount  of  work  involved  in  the  computation  of  the 
optical  system  of  a  typical  submarine  periscope.  Altogether 
the  number  of  separate  operations  is  something  like  40,000, 
the  mere  recording  of  which  would  fill  a  book  of  some  250 
pages. 

Of  course  all  this  sort  of  work  is  entirely  outside  anything 
that  can  be  done  in  school,  but  if  a  teacher  himself  is  entirely 
ignorant  of  it,  how  can  he  help  making  his  subject  unreal, 
and  talking  about  it  in  a  foreign  tongue? 

The  Sign  Convention 

Many  of  the  difficulties  underlying  the  teaching  of  ele- 
mentary optics  in  the  past  have  arisen  because  teachers 
have  adopted  different  practices  in  the  use  of  signs.  The 
following  diagram  shows  the  sign  convention  that  has  been 


INCIDENT  LIGHT  ^ 


agreed  upon  by  the  principal  optical  authorities  in  the  country. 

Books  to  consult: 

1.  Optics,  W.  E.  Cross. 

2.  Light,  C.  G.  Vernon. 

3.  The  Theory  of  Light  (new  ed.;,  T.  Preston. 

4.  Introduction  to  the  Theory  of  Optics,  A.  Schuster. 

5.  Experimental  Optics,  C.  F.  C.  Searle. 

6.  Practical  Optics,  B.  K.  Johnson. 

7.  Theory  of  Optics,  P.  Drude  (trans,  by  Mann  and  Millikan). 

8.  Optics,  Muller-Pouillet,  3rd  ed. 


GEOMETRICAL  OPTICS  517 

9.  Principles  and  Methods  of  Geometrical  Optics,  J.  P.  C.  Southall. 

10.  Optical  Measuring  Instruments,  Prof.  L.  C.  Martin. 

11.  Optical  Designing  and  Computing,  Prof.  Conrady. 

12.  Proceedings  of  the  Physical  Society,  No.  229;   the  papers  by 
Mr.  T.  Smith,  Dr.  Searle,  Dr.  Drysdale,  Mr.  C.  G.  Vernon,  Captain 
T.  Y.  Baker,  are  all  very  instructive. 

The  reader  may  usefully  refer  to  the  memorandum  prepared,  in 
January,  1931,  by  the  Council  of  the  British  Optical  Instrument 
Manufacturers'  Association.  The  facts  adduced  definitely  establish 
the  pre-eminence  of  the  British  position  in  the  optical  industry. 
The  tests  effected  in  the  National  Physical  Laboratory  are  alone 
sufficient  to  make  that  clear. 


CHAPTER  XXXV 

Map  Projection 

Developable  and  non -Developable  Surfaces 

It  is  the  geography  teacher's  business  to  show  how  maps 
can  be  outlined  on  the  particular  graticule  system  prepared 
for  him  by  the  mathematician.  This  graticule  system — a 
gridiron  or  lattice-work  system  of  parallels  and  meridians — 
is  in  its  very  essence  mathematical  and  should  be  included 
in  every  school  mathematical  course. 

Fundamental  principles  of  projection  will  already  have 
been  taught  in  the  lessons  on  geometry.  The  principles  of 
orthographic  projection,  including  so-called  "  plans  and 
elevations  ",  should  have  been  taught  thoroughly.  It  is 
just  an  affair  of  parallels  and  perpendiculars,  and  thence  to 
the  idea  of  parallel  rays  of  light  from  an  indefinitely  distant 
source  is  but  a  step. 

The  geometry  of  the  sphere  should  also  be  known 
thoroughly;  for  instance,  that  the  area  of  a  circle  is  ?rR2; 
of  a  sphere,  TrD2  and  therefore  4  times  one  of  its  great 


518  CRAFTSMANSHIP  IN  MATHEMATICS 

circles;  of  a  hemisphere,  twice  that  of  its  great  circle;  and 
that  the  volume  of  a  sphere  is  7rD3/6. 

It  should  be  realized  that  when  we  look  at  a  sphere  we 
cannot  see  the  whole  of  a  half  of  it.  The  portion  of  the 
visible  surface  is  that  encircled  by  a  tangent  cone  with  its 
apex  at  the  eye  (we  neglect  binocular  vision).  A  photograph 
of  a  geographical  globe  would  necessarily  give  a  picture  of 
rather  Jess  than  a  hemisphere. 

Developable  surfaces  is  another  subject  that  should  have 
been  taught.  A  paper  model  of  a  cube,  prism,  or  pyramid 
can  be  slit  open  along  some  of  its  edges  and  laid  out  on  the 
flat,  in  other  words,  "  developed  ".  A  cylinder  or  cone 
can  be  similarly  treated.  On  a  cylinder  or  cone  straight  lines 
can  be  drawn  in  certain  directions;  if  the  cylinder  or  the 
cone  is  lying  on  the  table,  the  line  of  contact  with  the  table 
is  one  such  straight  line.  But  a  spherical  surface  is  altogether 
of  a  different  type;  no  straight  line  can  be  drawn  upon  it; 
it  cannot  be  developed.  A  sphere  touches  a  plane  at  a 
point.  We  cannot  cover  a  sphere  with  a  sheet  of  paper  as 
we  can  a  cylinder  or  cone. 

Now  the  earth  is  approximately  spherical,  and  any  correctly 
drawn  map  is  part  of  that  spherical  surface.  An  atlas  of 
true  maps  would  consist  of  spherical  segments,  not  flat 
sheets.  Such  an  atlas  has  been  made  in  metal,  but  it  is  clumsy 
to  use  and  is  expensive.  For  convenience  we  draw  our  maps 
on  the  flat,  and  thus  they  are  all  wrong.  A  map  of  England 
drawn  to  scale  on  the  surface  of  an  orange  would  be  very 
small  but  large  enough  for  a  needle  to  be  thrust  through 
the  orange  along  a  chord  from  Bournemouth  to  Berwick. 
A  perfectly  straight  tunnel  driven  between  these  towns 
would  pass  under  Birmingham  4  miles  below  the  surface. 
If  a  map  of  Europe  be  sketched  to  scale  on  a  hollow  india- 
rubber  ball,  and  that  portion  of  the  ball  be  cut  out,  the  portion 
has  to  be  stretched  a  great  deal  to  lie  flat,  and  thus  parts 
of  the  map  are  greatly  distorted. 

Evidently  no  map  can  be  drawn  on  a  flat  surface  accurately. 
How  do  map-makers  set  to  work? 


MAP  PROJECTION  519 

If  we  examine  an  ordinary  geographical  globe,  we  see 
the  equator,  the  north  and  south  poles,  meridians  of  longitude 
running  from  pole  to  pole,  and  diminishing  circles  of  latitude 
running  "  parallel  "  to  the  equator.  And  on  this  network  of 
lines  we  see  a  true  map  of  the  world. 

To  draw  a  map,  we  first  draw  a  network  of  lines  corre- 
sponding as  nearly  as  possible  to  those  on  the  surface  of  the 
globe,  though  they  are  bound  to  differ  very  considerably 
from  the  originals.  The  network  once  drawn,  we  put  into 
each  little  compartment,  as  accurately  as  we  can,  the  corre- 
sponding bit  of  map  on  the  globe.  The  real  trouble  is  to 
draw  the  network. 

An  examination  of  an  atlas  shows  that  the  various  net- 
works differ  much  in  appearance.  Sometimes  one  or  both 
sets  of  lines  are  straight,  sometimes  curved,  and  the  cur- 
vature seems  to  vary  in  all  sorts  of  ways.  Why?  This  we 
must  try  to  find  out. 

In  an  ordinary  plan  drawn  to  scale,  say  of  a  house  or  of 
a  town,  we  have  the  simplest  form  of  projection,  called  the 
"  orthographic  ".  To  every  point  in  the  original  corresponds 
a  definite  point  in  the  drawing,  and  the  spatial  relations 
between  the  points  are  faithfully  reproduced;  only  the  scale 
is  changed. 

But  in  a  map,  the  relations  may  all  be  changed.  There 
will,  however,  still  be  a  systematic  one-to-one  correspondence 
of  points.  Some  sort  of  general  resemblance  to  the  original 
may  always  be  easily  detected,  though  there  is  certain  to  be 
distortion  of  form,  or  inequality  in  area,  or  both. 

The  map-maker  is  bound  to  sacrifice  something.  If  he 
is  making  a  map  for  teaching  geography,  he  tries  to  represent 
correctly  the  relative  sizes  of  land  and  sea  areas  and  thus 
provides  an  equal-area  projection.  If  he  is  making  a  map  for 
a  navigator,  he  tries  to  show  correct  directions,  and  does 
not  trouble  much  about  size.  Or  he  may  be  concerned 
mainly  with  correct  shapes,  and  not  much  with  sizes  and 
directions.  Hence  he  has  contrived  projections  for  different 
purposes.  He  has  to  be  content  to  represent  a  portion  of 


520  CRAFTSMANSHIP  IN  MATHEMATICS 

the  earth's  surface  accurately  in  certain  respects  and  to 
let  other  considerations  go. 

The  plan  adopted  is  to  project  the  curved  lines  from  the 
globe  on  to  (1)  a  plane  surface,  or  (2)  a  developable  surface 
(cylinder  or  cone). 

Some  projections  are  readily  effected  geometrically;  they 
are  easy  to  draw  and  to  understand.  Other  projections  are 
not  strictly  geometrical:  they  are  compromises,  effected  for 
some  particular  purpose,  and  are  often  called  transformations. 
In  these  cases  point-to-point  correspondence  is  determined 
merely  by  formulae  which  express  the  position  of  each  point 
on  the  plane  of  the  projection  in  terms  of  the  position  of  the 
point  on  the  spherical  surface  to  which  it  corresponds. 

Projection  Shadows 

It  is  possible  to  obtain  geometrical  projections  by  casting 
shadows.  A  light  is  placed  in  a  suitable  position,  and  a  pencil 
outline  of  the  shadow  of  the  globe  is  traced  on  a  conveniently 
placed  plane.  This  done,  it  is  easy  to  see  how  a  better  pro- 
jection may  be  made  with  ruler  and  compasses. 

Of  course  if  we  use  a  solid  globe  the  shadow  will  be 
merely  a  black  circle.  We  require  a  hollow  translucent 
globe,  with  the  meridians  and  parallels  painted  black  on  the 
surface,  and  a  strong  light  inside.  If  the  globe  is  fixed  near 
a  sheet  of  white  paper  on  the  wall  or  on  the  table,  the  shadows 
of  some  of  the  meridians  and  parallels  will  be  cast  on  the 
paper,  and  those  fairly  near  the  globe  will  be  clear  enough 
to  be  pencilled  over.  A  large  white  porcelain  globe  used  for 
gas  and  electric  lighting  answers  the  purpose  well. 

When  teaching  40  years  ago,  I  found  that  a  better  plan 
was  to  use  a  spherical  wire  cage  instead  of  a  globe,  made 
something  after  the  pattern  of  the  old-fashioned  wire  pro- 
tectors of  naked  gas-flames  in  factories.  Such  a  cage  2'  or 
2'  6"  in  diameter  is  easily  made  in  the  school  workshop. 
It  is  merely  a  question  of  bending  wire  and  soldering  a  number 
of  joints.  For  the  equator,  a  rather  stouter  wire  should  be 


MAP  PROJECTION 


521 


used  than  for  the  other  circles.  The  meridians  are  best 
not  made  of  complete  circles  but  of  rather  less  than  half 
circles,  fastened  into  a  ring  4"  or  5"  in  diameter,  after  the 
manner  of  the  ribs  at  the  top  of  an  umbrella.  It  is  true  that 
the  actual  north  and  south  poles  will  be  missing  but  this 
cannot  be  helped,  the  crossing  of  12  wire  circles  at  a  common 
point  not  being  practicable.  The  meridians  and  parallels 
may  be  placed  at  15°  intervals.  Two  half-cages  are  also 
desirable,  one  with  a  pole  at  its  centre,  one  with  a  point  on 


Fig.  275 

the  equator  at  its  centre.  The  three  should  be  mounted  on 
suitable  stands,  in  order  that,  in  use,  they  may  easily  be 
kept  in  a  fixed  position. 

The  main  difficulty  is  the  provision  of  a  suitable  light. 
Theoretically  we  require  the  light  to  be  concentrated  at  a 
point.  As  this  is  impossible,  we  use  a  small  electric  bulb, 
porcelain  or  similar  material,  with  the  most  powerful  light 
obtainable.  A  darkened  room  is,  of  course,  necessary. 


Main  Types  of  Projection 

The   principal   projections   may    be    grouped   under  six 

main   heads:    (1)    zenithal   or   azimuthal;    (2)    globular;  (3) 

conical;    (4)   cylindrical;    (5)  sinusoidal;    (6)  elliptical.  Of 
most  of  these  there  are  various  modifications. 


522  CRAFTSMANSHIP  IN  MATHEMATICS 

(1)  Zenithal  or  Azimuthal  Projection 

This  projection   derives  its    two   names   from  the   facts 

(1)  the  map  is  symmetrical  about  its  central  point,  just  as 
the  stellar  vault  is  symmetrical  about  the  zenith  of  the  observer; 

(2)  the  projection  preserves  the  azirmiths  of  distances  measured 
from  the  map's  centre. 

There   are   three   distinct  types   of  this   projection:    (1) 
orthographic;   (2)  stereographies   (3)  gnomonic.     (See  fig.  276.) 

1.  Orthographic. — This  is  simply  an  affair  of  perpen- 
diculars and  parallels.  As  we  cannot  obtain  parallel  rays 
by  artificial  light,  we  must  use  the  sunlight  at  some  con- 
venient hour.  Let  the  paper  prepared  to  receive  the  shadow 
be  placed  at  right  angles  to  the  direction  of  the  sun's  rays. 
Fix  the  skeleton  wire  hemisphere  (ii)  so  that  the  equator 
is  parallel  to  the  paper;  the  parallels  of  latitude  will  be 
projected  as  circles  of  true  size,  the  meridians  as  radii  of 
these  circles.  Geometrically,  we  draw  the  projection  exactly 
as  in  geometry.  Note  that  the  scale  along  the  circles  is  always 
true,  but  that  the  radial  lines  are  foreshortened  more  and 
more  as  the  distance  from  the  centre  increases. 
,  2.  Stereographic. — Set  up  the  same  wire  hemisphere, 
with  its  equatorial  plane  vertical  and  parallel  to  the  projection 
plane,  and  place  the  lamp  at  the  further  extremity  of  the 
diameter  corresponding  to  the  earth's  axis,  that  is  at  the 
"  south  pole  ".  The  parallels  of  latitude  are  again  projected 
as  circles,  but  they  are  enlarged,  the  equator  being  twice 
the  size  of  the  original.  The  meridians  are  projected  as 
radii,  as  before. 

This  projection  has  a  general  similarity  to  the  orthographic, 
and  its  geometrical  construction  is  a  useful  exercise.  The 
scale  is  increased  equally  along  the  meridians  and  parallels, 
and  some  good  Sixth  Form  problems  may  be  based  on  the 
projection.  In  particular,  the  projection  provides  a  ready  means 
of  studying  the  sum  of  the  angles  of  a  spherical  triangle. 

3.  Gnomonic. — For  this  projection,  the  light  is  placed  at 
the  centre  of  the  sphere.  Although  the  parallels  of  latitude  are 


MAP  PROJECTION 


523 


o 

o 

o 

z 
o 


O 
u 
a: 
u 

fe 


o 

x 
a 


o 

i 

oc 
O 


524 


CRAFTSMANSHIP  IN  MATHEMATICS 


still  projected  as  true  circles,  they  are  still  more  enlarged, 
and  the  equator  itself,  having  the  light  in  its  own  plane, 
cannot  be  projected  at  all.  The  general  appearance  of  the 
projection  is  similar  to  that  of  the  other  two,  but  obviously 
the  areas  very  far  from  the  pole  are  greatly  distorted  in  the 
projection.  The  figure  shows  the  polar  region  to  within 
30°  of  the  equator.  It  makes  a  fairly  good  map  for  areas 
within  30°  of  the  pole. 

The  three  projections  may  be  usefully  compared  in  this 
way: 

N M   P     Q 


Let  NESQ'  be  a  meridian  of  the  earth,  and  XY  its  pro- 
jection plane.  Take  a  point  L  at  polar  distance  a.  Then 
angle  LSN  —  |a.  Let  radius  be  R. 

Orthographic  projection  of  arc  NL  =  NM  =  R  sin  a. 
Stereographic          „  „  =  NP    =  2R  tan^a. 

Gnomonic  „  „  =  NQ  =  R  tana. 


Observe  that  in  the  orthographic  projection  the  outer 
circles  are  crowded  together,  in  the  Stereographic  the  outer 
are  farther  apart  than  the  inner,  and  in  the  gnomonic  the 
outer  circles  get  so  far  apart  as  to  be  useless.  It  is  some- 
times convenient  to  arrange  these  circles  at  equal  dis- 


MAP  PROJECTION 


525 


tances  apart,  and  then  we  have  the  zenithal  equidistant  pro- 
jection. It  is  also  possible  for  the  distances  of  the  parallels 
of  latitude  so  to  be  regulated  that  the  area  enclosed  by  any 
parallel  is  equal  to  the  area  of  the  globe  cut  off  by  the  same 
parallel,  and  then  we  have  the  zenithal  equal-area  projection. 
Strictly,  these  are  not  true  projections,  but  the  associated 
geometry  is  interesting  and  instructive. 


GO; 


(2)  Globular  Projection 

All  three  zenithal  projections  are  sometimes  called  "  per- 
spective "  projections,  since  they  can  be  cast  as  shadows. 
But    the    globular    projection 
cannot   be   cast   as   a  shadow, 
and  is  therefore  non-perspec- 
tive. 

The  geometry  is  a  useful 
exercise  for  beginners.  The 
projection  is  commonly  used 
for  maps  of  the  world  in  two 
hemispheres.  The  figure  repre- 
sents one  hemisphere.  Divide 
the  equatorial  diameter  into 
an  equal  number  of  parts,  say 
parts  representing  30°.  Divide 
the  circumference  similarly. 
The  curves  are  all  arcs  of  circles,  each  to  be  drawn  through 
three  points.  The  mathematics  of  the  projection  is  of  the 
simplest. 

(3)  Conical  Projection 

For  this  we  require  the  wire  skeleton  of  the  complete 
globe,  with  the  light  fixed  at  the  centre.  The  shadow  will 
be  cast,  not  on  a  plane,  but  on  the  inner  surface  of  a  white 
paper  cone. 

Fold  up,  in  the  usual  way,  a  common  filter  paper,  and 


526 


CRAFTSMANSHIP  IN  MATHEMATICS 


fit  it  into  a  funnel.  It  makes  a  cone  with  a  60°  apex.  A  half 
circle  of  paper  would  make  the  same  cone,  the  two  halves 
of  the  diameter  being  brought  together.  A  sector  having 
an  apex  of  less  than  180°  folds  up  into  a  more  pointed  cone; 
one  with  an  apex  of  more  than  180°  folds  up  into  a  flatter 
cone.  A  sector  of  360°  (a  complete  circle)  necessarily  remains 
a  plane. 

Make  a  white  paper  cone  (of  about  130°  apex  in  the  flat), 
slip  it  over  the  polar  region  of  the  skeleton  globe  so  that  the 
apex  is  in  a  line  with  the  axis  of  the  globe.  The  cone  touches 
the  sphere  tangentially,  viz.  in  a  circle,  and  this  circle  is  a 
parallel  of  latitude.  If  this  corresponds  with  one  of  the  wire 


Fig.  279 

circles  so  much  the  better.  Now  gently  mark  in  the  out- 
lines of  the  cast  shadow.  This  is  pretty  easy  in  the  neigh- 
bourhood of  the  line  just  mentioned,  but  the  cast  shadow 
gets  very  faint  as  we  get  farther  away  from  the  line.  Now  open 
out  the  cone  on  the  flat  (fig.  279,  ii),  and  we  have  an  ordinary 
conical  projection.  The  arc  represented  by  a  heavy  line 
RSR'  is  the  circle  of  contact  RS  in  (i),  the  "  standard  parallel  ", 
and  it  is  divided  exactly  as  the  circle  it  touches  on  the  sphere 
is  divided. 

The  solid  angle  N  of  the  sphere  =  4  right  angles.  The 
angle  of  the  cone  when  developed  is  angle  RP'R'.  The 
ratio  of  the  latter  angle  to  the  former  is  called  the  constant 


MAP  PROJECTION 


527 


Fig.  280 


of  the  cone.  It  is  a  simple  Fifth  Form  problem  to  prove  that 
this  constant  is  the  sine  of  the  latitude  of  the  standard 
parallel. 

The  geometrical  construction  is  simple.  Observe  that 
the  parallels  are  arcs  of  circles,  and  that  the  meridians  are 
straight  lines.  Since  meridians  are  great  circles  and  their 
planes  pass  through  the  centre 
of  the  globe,  these  planes  must 
bisect  the  cone  and  therefore 
cut  its  surface  in  straight  lines. 
The  projection  is  commonly 
used  for  countries  in  middle 
latitudes  if  the  latitude  is  not 
of  too  great  an  extent,  e.g.  for 
England.  The  conical  projec- 
tion with  two  standard  parallels 
(fig.  280)  is  a  common  projection 
for  the  larger  European  coun- 
tries. Its  principle  is  equally 
simple. 

(4)  Cylindrical  Projection 

Take  a  large  sheet  of  white  paper  and  convert  it  into  a 
cylinder  of  the  same  diameter  as  the  skeleton  wire  sphere. 
Its  length  should  be  3  or  4  times  the  diameter.  Slip  it  over 
the  sphere  so  that  the  equator  is  in  about  mid-position, 
and  place  the  light  at  the  centre  of  the  sphere.  A  shadow 
of  a  part  of  the  wire  sphere  is  cast  on  the  cylinder.  Obviously 
the  shadows  of  the  two  poles  cannot  be  cast  on  the  cylinder 
at  all,  and  high  latitudes  are  cast  at  great  distances,  with 
consequently  great  distortion.  The  small  circle  of  latitude 
AB  will  appear  as  A'B';  in  fact  all  circles  of  latitude  will 
be  projected  as  circles  on  the  cylinder  and  will  all  be  of  the 
same  size  as  the  equator.  All  meridians,  being  great  circles, 
will  be  cast  as  straight  lines.  Open  out  the  cylinder  on  the 
flat  (ii),  and  the  projection  is  seen  to  consist  of  a  net  of  rect- 
angles. E'Q'  =  TrEQ  and  may  be  subdivided  in  the  usual  way 


528 


CRAFTSMANSHIP  IN  MATHEMATICS 


The  projection  is  not  of  much  practical  value.  Except 
in  the  immediate  neighbourhood  of  the  equator  there  is  far 
too  much  distortion. 


Fig.  281 

But  various  modifications  of  this  primary  cylindrical 
projection  have  been  adopted,  two  of  them  being  noteworthy: 
(1)  Lambert's  equal-area  projection,  and  (2)  Mercator's 
projection. 

1.  Lambert's  projection. — Construction:  divide  the  quadrant 


Fig.  282 


NQ  (fig.  282)  into  an  equal  number  of  parts,  say  6  of  15° 
each,  and  draw  parallels  to  EQ,  and  so  obtain  parallels  of 
latitude.  For  meridians,  make  E'Q'  —  TrEQ,  and  divide 
up  into  intervals  of,  say,  30°.  Note  that  the  parallels  of  lati- 


MAP  PROJECTION  529 

tude  are  horizontal  lines  at  a  distance  of  r  sin  A  from  the 
equator  (A  =  lat.). 

It  is  a  well-known  theorem  in  geometry  that  the  area 
between  any  two  parallels  on  the  enveloping  cylinder  is 
equal  to  that  of  the  corresponding  zone  on  the  globe.  Hence 
the  area  of  the  rectangle  MM'  is  equal  to  the  area  of  the  globe. 
The  proof  of  the  theorem  should  be  given. 

2.  Mercator's  orthomorphic  projection. — This  is  the  best- 
known  of  all  projections;  it  is  used  for  navigation  purposes, 
and  for  maps  of  the  world.  But  it  is  responsible  for  many 
geographical  misconceptions,  for  instance  the  misleading 
appearance  of  the  polar  areas,  which  are  greatly  exaggerated. 
Greenland  is  made  to  appear  larger  than  South  America, 
though  only  one-tenth  its  size. 

As  with  all  cylindrical  projections,  the  meridians  are 
equidistant  parallel  lines;  the  parallels  of  latitude,  on  the 
other  hand,  increase  in  distance  from  one  another  the  farther 
they  are  from  the  equator.  This  spacing  of  the  parallels 
of  latitudes  is  so  arranged  that  at  any  point  of  intersection 
of  parallels  and  meridians  (in  practice,  any  small  area),  the 
scale  in  all  directions  is  the  same.  Hence  the  projection  is 
orthomorphic.  Literally  the  term  means  "  preserving  the 
correct  shape  ". 

The  essential  characteristic  of  the  projection,  then,  is  this 
— that  at  any  point  the  scale  along  meridian  and  parallel  is 
the  same.  We  give  Dr.  W.  Garnett's  ingenious  illustration 
of  the  method  of  effecting  this. 

Dr.  Garnett  takes  a  very  narrow  gore,  i.e.  a  strip  between 
two  meridians  on  the  globe  (cf.  the  surface  of  a  natural 
division  of  an  orange,  selected  for  its  narrowness),  and 
spreads  it  out  as  flat  as  possible;  if  very  narrow  there  is  no 
great  difficulty  in  spreading  it  out  very  nearly  flat,  without 
much  distortion;  then  it  is  very  nearly  an  equal-area  strip, 
i.e.  its  area  on  the  flat  is  very  nearly  the  same  as  when  it  was 
part  of  the  curved  surface  of  the  sphere.  The  length  of  the 
spread-out  gore  is,  of  course,  half  the  circumference  of  the 
orange. 

(E291  35 


530 


CRAFTSMANSHIP  IN  MATHEMATICS 


Let  NAB  represent  the  half  gore,  AB  representing  10° 
at  the  equator;  and  let  NM  be  the  central  meridian. 
Divide  NM  into  9  equal  parts,  and  through  the  points  of 
division  draw  the  parallels  shown  in  the  figure;  these  re- 
present 10°  intervals  of  latitude  from  the  equator  to  the 
pole.  Suppose  the  gore  to  be  made  of  malleable  metal. 


30'     40'      60 


Fig.  283  * 


Hammer  it  out  in  such  a  way  as  to  cause  it  to  spread  to  the 
uniform  width  AB.  Clearly  we  cannot  do  this  in  the  im- 
mediate neighbourhood  of  N:  there  would  not  be  enough 
metal.  Hence  cut  the  gore  off  at  about  85°.  But  the  gore 
cannot  be  hammered  out  without  expanding  in  length  as 
well  as  in  breadth.  At,  say,  40°  little  hammering  will  be 
required,  and  the  additional  length  there  will  be  slight; 
but  at,  say,  70°  much  hammering  will  be  required  to  produce 


Fig.  284  • 

the  necessary  additional  width,  and  therefore  there  will  be 
much  additional  length  produced.  At  45°  the  ratio  of  the 
increased  width  to  the  original  width  is  \/2  :  1,  and  therefore 
the  length  of  the  strip  at  45°  is  increased  \/2  times,  and  hence 
the  area  is  increased  there  <\/2  X  1/2  times,  that  is,  twice, 
and  the  thickness  is  therefore  halved.  At  60°  there  will  be  a 
doubling  of  width  and  therefore  a  doubling  of  length,  i.e. 

*  Figs.  283  and  284  are  made  to  lie  down,  to  save  space.    Normally,  the  gores 
would  be  given  an  upright  position. 


MAP  PROJECTION  531 

the  area  will  be  multiplied  by  4  and  the  thickness  reduced 
toj. 

It  is  easy  to  imagine  the  whole  series  of  36  gores  (fig.  284 
shows  4)  placed  side  by  side,  and  rolled  out  until  the  edges 
meet  and  36  rectangles  are  formed. 

Generally,  every  little  strip  parallel  to  the  equator  is 
increased  both  in  length  and  breadth  in  proportion  as  the 
radius  of  the  sphere  is  to  the  radius  of  the  circle  of  latitude 
where  the  strip  is  situated.  At  80°  the  area  is  increased 
about  33  times,  and  at  85°  about  132  times.  The  figure 
(fig.  283)  shows  roughly  how  the  gore  between  0°  and  80° 
is  hammered  out  into  the  rectangle  ABPQ. 

If  the  36  gores  were  extended  to  lat.  80°  N.  and  S.,  and 
placed  side  by  side,  we  should  have  a  rectangle  36  times 
AB  in  length  and  twice  AP  in  height,  and  we  should  have  the 
framework  for  a  Mcrcator  map  of  the  world  between  the 
parallels  80°  N.  and  80°  S. 

The  point  about  the  whole  projection  is  the  retention  of 
true  shape,  though  this  applies  to  only  very  small  areas. 
At  the  equator,  areas  are  unchanged;  at  80°  they  are  in- 
creased 33  times. 

The  shapes  of  small  areas  are  magnified,  not  distorted. 
Strictly  the  orthomorphism  is  applicable  only  to  points  and 
is  therefore  only  theoretical. 

Construction  of  a  M  creator  map.  —  The  radius  of  a  parallel 
of  latitude  on  a  sphere  of  radius  r  is  r  cos#.  Hence  if  a  degree 
of  longitude  in  latitude  0  is  to  be  made  equal  to  a  degree 
at  the  equator,  its  length  must  be  divided  by  cos0.  If  the 
scale  of  the  map  is  to  be  increased  in  all  directions  in  the 
same  ratio,  then  the  length  of  the  degree  of  latitude  measured 
along  the  meridian  must  also  be  increased  in  the  same  ratio. 
If  y  be  the  distance  of  the  parallel  of  latitude  0  from  the 
equator  in  the  Mercator  map  of  a  sphere  of  radius  r, 

e 


a  formula  which  may  be  evaluated  by  Sixth  Form  boys. 


532  CRAFTSMANSHIP  IN  MATHEMATICS 

The  distances  of  the  parallels  from  the  equator  are,  in 
terms  of  the  radius,  approximately,  for 


10° 

•176  R 

50° 

1-011  R 

20° 

•356  R 

60° 

1-317R 

30° 

•55  R 

70° 

1-736  R 

40° 

•763  R 

80° 

2-436  R 

These  values  should  be  checked  from  a  Mercator  in  a  good 
atlas:  equator  =  27T.R. 

Mercator,  and  Great  Circle  Sailing. — The  special  merit 
of  Mercator's  projection  lies  in  the  fact  that  any  given  uniform 
compass  course  is  represented  by  a  straight  line.  All  meridians 
are  exactly  north  and  south,  and  all  parallels  exactly  east 
and  west.  Hence  a  navigator  has  only  to  draw  a  straight 
line  between  his  two  ports,  and  the  angle  this  line  makes 
with  the  meridian  on  the  map  gives  his  true  course  for  the 
whole  voyage. 

Any  straight  line  drawn  in  any  direction  on  a  Mercator 
is  called  a  rhumb  line\  it  crosses  all  parallels  at  a  constant 
angle,  and  all  meridians  similarly.  A  sailor  who  is  told  to 
sail  on  a  constant  bearing  simply  sets  his  compass  according 
to  the  rhumb  line. 

But  this  course  may  not  be  the  shortest;  it  cannot  be, 
unless  it  is  along  the  equator  or  along  a  meridian,  i.e.  along 
a  great  circle.  A  rhumb  course  in  any  other  direction  is  not 
along  a  great  circle,  and  we  know  that  the  shortest  distance 
between  two  points  in  a  sphere  is  along  the  great  circle 
passing  through  them.  Economy  makes  the  navigator  take 
the  shortest  course  if  he  can.  How  is  he  to  find  it? 

A  rough  and  ready  way  would  be  to  take  a  wire  hoop 
that  would  exactly  fit  round  the  equator  or  round  one  of  the 
meridians  (and  therefore  round  a  great  circle:  we  neglect 
the  ellipticity  of  the  earth),  hold  it  over  the  globe  so  that  it 
passed  through  the  two  ports  at  the  ends  of  the  course  under 
consideration,  chalk  in  the  curve,  and  then  transfer  the 
curve  to  the  Mercator,  freehand,  as  accurately  as  the  corre- 
sponding graticules  would  allow. 


MAP  PROJECTION 


533 


A  navigator  always  follows  a  great  circle  if  he  can,  not 
the  rhumb  line,  and  for  his  special  use  great  circle  courses  are 
calculated  and  laid  down  on  a  Mercator's  chart. 

If  ARE  is  the  rhumb  line  between  two  places  A  and  B 
(the  figure  is  a  fragment  of  a  Mercator  chart),  and  AGCB  is 
the  great  circle  (and  therefore  shorter  than  the  rhumb  line), 
a  navigator  might  sail  along  a  series  of  chords  AG,  GC,  CB, 
altering  his  course  at  G  and  C.  He  would  not  quite  follow 


Fig.  285 


the  great  circle,  but  he  would  follow  a  much  shorter  route 
than  the  rhumb  line  course. 

Give  the  boys  examples  of  the  course  between,  say, 
Japan  and  Cape  Horn,  Plymouth  and  New  Orleans,  Cape 
Town  and  Adelaide.  Let  them  mark  in  roughly  both  the 
rhumb  line  and  the  great  circle  courses  on  a  Mercator  chart. 
Remind  them  of  the  deceptive  geometry,  as  in  fig.  285,  where 
the  chord  represents  a  longer  distance  than  the  arc  it  subtends. 

To  trace  the  course  of  a  great  circle  on  a  Mercator  chart. — 
Any  great  circle  must  cut  the  equator  at  two  places  and  at 
a  given  angle.  Hence  it  will  cut  (i)  a  given  meridian  at  a 
point  whose  latitude  can  be  determined,  and  (ii)  a  given 
parallel  of  latitude  at  a  point  whose  longitude  can  be  de- 
termined. 

Assume  that  we  are  given: 

(i)  a,  the  inclination  of  the  great  circle  to  the  plane  of 

the  equator; 
(ii)  A,  the  longitude,  measured  from  one  of  the  points 

of  section,  of  a  meridian  in  latitude  L. 


534  CRAFTSMANSHIP  IN  MATHEMATICS 

Then  the  following  equation  may  be  established: 

tanL  =  tana  .  sin  A, 
or,     sin  A  —  cot  a  .tanL. 

From  this  equation,  either  the  latitude  can  be  determined  at 
which  the  great  circle  cuts  any  meridian,  or  the  longitude 
at  which  it  cuts  any  parallel.  The  equation  may  therefore 
be  used  to  trace  the  course  of  a  great  circle  on  a  Mercator 
chart. — Sixth  Form  boys  should  work  through  a  few  of  the 
exercises  in  Nunn,  Exercises,  Vol.  II. 

Aviators  are  naturally  much  interested  in  great  circle 
sailing.  Let  the  boys  determine  an  aviator's  route  between 
two  given  places,  say  5000  miles  apart,  by  stretching  a  string 
over  a  geographical  globe.  Then  ask  them  how  an  aviator 
would  set  his  compass.  Let  them  lay  down  the  course  on 
a  Mercator  chart  (graphically  and  approximately  will  do), 
and  see  how  it  differs  from  the  rhumb  line,  and  how  compass 
directions  might  be  determined  by  a  succession  of  chords. 

(5)  Sinusoidal  Equal -area  Projection 

This  is  sometimes  called  the  Sanson  Flamsteed  projection; 
it  is  used  mainly  for  world  maps.  An  equal-area  or  "  homo- 
lographic  "  projection  is  a  projection  where  shape  is  sacrificed 
to  equality  of  area. 

It  differs  widely  from  the  geometrical  and  (mainly) 
shadow  projections  already  considered. 

The  equator  (=  2?rR)  is  true  to  scale.  The  central  meridian 
(—  TrR)  is  also  true  to  scale.  Parallels  of  latitude  are  equidistant 
horizontal  lines.  All  the  meridians  are  of  the  form  of  sine 
curves.  Each  parallel  is  equally  divided  by  the  meridians, 
which  are  nearer  and  nearer  together  towards  the  poles. 

Fig.  286  shows  a  quarter  of  the  complete  projection  of  the 
world  map;  EZ  =  2NZ.  Divide  NZ  into,  say,  6  equal  parts 
(of  15°  each),  and  EZ  into  6  parts  (of  30°  each).  Each  hori- 
zontal straight  line  is  equal  in  length  to  the  corresponding 


MAP  PROJECTION 


535 


circle  of  latitude.  Through  the  extremities  of  these  lines 
draw  the  curve  EN  which  represents  the  boundary  of  the 
quarter  map.  Divide  every  parallel  into  6  equal  parts,  similar 
to  EZ,  and  draw  curves  through  the  corresponding  points 

N 


60 


30" 


71 


180° 


60' 


Fig.  286 


of  division;  these  curves  are  meridians  (in  the  figure,  half- 
meridians). 

The  disadvantage  of  the  projection  is  that  towards  the 
edge  the  meridians  are  very  oblique  and  thus  the  shape  is 
much  distorted.  The  graticules  along  the  equator  and  central 
meridian,  on  the  other  hand,  practically  retain  their  original 
shape. 

In  any  projection  graticule,  the  horizontal  lengths  are 
exactly  the  same  as  in  the  graticule  on  the  globe;  and  the 
vertical  height  of  the  projection  graticule  is  equal  to  the 
length  of  the  corresponding  piece  of  meridian  on  the  globe. 
Hence  the  area  of  any  projection  graticule  is  equal  to  the 
area  of  the  corresponding  graticule  on  the  globe,  or  the  whole 
area  of  the  map  is  equal  to  the  whole  area  of  the  globe. 

Each  curved  meridian  is  a  sine  curve:  why?  Might  the 
sine  curves  be  drawn  before  the  parallels? 

The  projection  is  very  good  for  maps  of  Africa  and  South 
America.  Why? 


536 


CRAFTSMANSHIP  IN  MATHEMATICS 


(6)  Mollweide's  Elliptical  Projection 

This  projection  is  also  used  for  world  maps.  Again  the 
parallels  are  horizontal  lines.  The  meridians  are  ellipses 
(there  are  two  special  cases:  the  central  meridian  is  a  straight 
line  and  the  90°  meridian  is  a  circle). 

Again  the  area  of  the  map  is  equal  to  the  area  of  the  surface 
of  the  globe. 

Since  the  area  of  the  surface  of  the  sphere  is  equal  to 
4  times  the  area  of  its  great  circle,  the  area  of  the  hemisphere 
is  equal  to  twice  the  area  of  its  circular  base. 


Fig.  287 

Let  the  radius  of  the  globe,  fig.  287  (i),  be  R.  Draw  a  circle 
(ii)  of  radius  \/2.R  (=  CB).  Area  —  2?rR2  sq.  in.,  which  is  the 
area  of  the  half  globe.  Let  C  be  the  centre  of  the  circle; 
draw  a  horizontal  diameter  ACB  and  a  vertical  diameter 
NCS.  Produce  AB  so  that  CE  =  2CA  and  CQ  =  2CB. 
Draw  an  ellipse  having  EQ  and  NS  for  axes.  It  is  one  of  the 
properties  of  the  ellipse  that  if  any  line  KLMN'  be  drawn 
parallel  to  EQ  cutting  the  ellipse  and  the  circle,  KN'  =  2LM; 
and  as  this  is  true  for  any  such  line,  it  follows  that  the  area 
of  the  ellipse  =  twice  the  area  of  the  circle  =  the  area  of 
the  globe. 

Divide  EQ  into  equal  parts  and  through  the  points  of 
division  draw  ellipses  with  NS  as  a  common  axis;  these  are 


MAP  PROJECTION  537 

the  meridians.  Evidently  all  gores  (e.g.  NnSC,  NwSw)  are 
equal  in  area.  For  an  equal-area  projection,  it  remains  to 
divide  these  gores  by  parallels  of  latitude  into  the  same  areas 
as  the  corresponding  gores  between  the  meridians  on  the  globe 
are  divided.  This  is  the  only  difficult  part  of  the  problem. 

We  have  to  draw  KN'  so  that  it  will  correspond  to  some 
particular  degree  of  latitude  <f>  on  the  globe.  Fig.  287  (i)  repre- 
sents a  section  of  the  globe  through  the  great  circle  NA'SB'. 
In  fig.  (ii),  the  circle  represents  the  area  of  the  hemisphere 
and  the  ellipse  the  area  of  the  whole  sphere. 

The  area  of  the  zone  L'A'B'M'  on  the  spherical  surface 
(radius  =  R)  in  fig.  (i)  is  equal  to  twice  the  area  of  the  zone 
LABM  on  the  plane  surface  (radius  —  ^/2.R)  in  fig.  (ii). 

We  have  to  find  the  angle  MCB.    Let  it  equal  a.   Then 

2a  -f-  sin2a  =  TT  sin9. 

It  is  not  easy  from  this  equation  to  obtain  a  in  terms  of  <£, 
but  it  is  quite  easy  to  determine  cf>  in  terms  of  a.  Hence  if 
any  parallel  be  drawn  in  the  ellipse,  and  the  angle  a  is  measured, 
the  latitude  </>  to  which  it  corresponds  is  found  at  once. — 
The  formula  should  be  established  by  the  Sixth  Form. 

Choice  of  Projection 

Let  the  boys  examine  a  good  modern  atlas  in  which  the 
projections  used  are  named;  and  get  the  boys  to  discover 
why  a  particular  projection  is  used  in  each  case.  This  may 
give  rise  to  an  interesting  discussion. 

Books  to  consult: 

1.  A  Little  Book  on  Map  Projection,  Garnett. 

2.  The  Study  of  Map  Projection,  Steers. 

3.  Map  Projections,  Hinks. 


538  CRAFTSMANSHIP  IN  MATHEMATICS 

CHAPTER  XXXVI 

Statistics 

The  Importance  of  the  Subject 

It  is  highly  desirable  that  an  elementary  study  of  this 
subject  shall  be  included  in  any  Sixth  Form  course.  Statistics 
enter  largely  into  modern  science  and  administrative  practice, 
and  the  underlying  principles  have  now  been  so  well  worked 
out  and  have  become  so  definite,  that  no  large  office,  govern- 
ment or  local ,  can  afford  to  be  without  at  least  one  well- 
trained  statistician.  The  newer  developments  of  psychology 
depend  almost  entirely  upon  a  rational  interpretation  of 
statistics.  There  are  some  teachers  who  are  still  ignorant 
of  the  principles  underlying  the  correct  handling  of  the 
statistics  of  everyday  school  practice;  and  thus  they  are 
necessarily  unable  to  make  the  most  effective  use  of,  for 
instance,  an  ordinary  sheet  of  tabulated  examination  results. 

I  have  seen  the  subject  taken  up  seriously  in  only  two 
or  three  schools,  and  have  therefore  had  little  experience  of 
the  methods  of  teaching  it.  The  teaching  suggestions  in 
Professor  Nunn's  Algebra  are  recognized  as  the  most  practi- 
cable yet  made,  and  the  topics  he  selects  for  inclusion  in  a 
school  course  seem  to  be  just  about  right.  The  technical 
side  of  the  subject  is,  of  course,  rather  difficult  for  boys, 
but  the  fundamentals  are  easy  to  grasp,  and  it  is  possible 
to  map  out  an  excellent  preliminary  course  that  will  give  a 
good  general  insight  into  the  subject  and  into  its  methods. 

The  main  problems  to  be  considered  may  be  grouped 
under  the  three  usual  heads:  (1)  frequency  distribution  of 
a  series  of  measurements  or  other  statistics;  (2)  frequency 
calculation:  probability;  (3)  correlation. 


STATISTICS  539 

Frequency  Distribution 

Frequency  distribution  is  concerned  with  the  best  ways 
of  recording  statistics  and  of  expressing  most  simply  and 
effectively  the  information  which  they  contain.  Suppose, 
for  instance,  a  Local  Education  Authority  has  examined 
20,000  children  between  the  ages  of  10  years  6  months  and 
11  years  6  months  for  scholarships  to  be  held  in  the  local 
Secondary  Schools.  What  would  be  the  best  way  of  recording 
the  results,  so  that  not  only  might  the